# Limits, Continuity and Differentiability · Mathematics · JEE Main

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## Numerical

JEE Main 2024 (Online) 9th April Morning Shift
Let $$\lim _\limits{n \rightarrow \infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2 n}{\left(n^2+1\right) \sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8 n}{\... JEE Main 2024 (Online) 9th April Morning Shift Let$$f:(0, \pi) \rightarrow \mathbf{R}$$be a function given by$$f(x)=\left\{\begin{array}{cc}\left(\frac{8}{7}\right)^{\frac{\tan 8 x}{\tan 7 x}}, ...
JEE Main 2024 (Online) 1st February Morning Shift
Let $\{x\}$ denote the fractional part of $x$ and $f(x)=\frac{\cos ^{-1}\left(1-\{x\}^2\right) \sin ^{-1}(1-\{x\})}{\{x\}-\{x\}^3}, x \neq 0$. If $\ma... JEE Main 2024 (Online) 31st January Evening Shift If $$\lim _\limits{x \rightarrow 0} \frac{a x^2 e^x-b \log _e(1+x)+c x e^{-x}}{x^2 \sin x}=1$$, then $$16\left(a^2+b^2+c^2\right)$$ is equal to ______... JEE Main 2024 (Online) 30th January Morning Shift If the function $$f(x)= \begin{cases}\frac{1}{|x|}, & |x| \geqslant 2 \\ \mathrm{a} x^2+2 \mathrm{~b}, & |x| is differentiable on$$\mathbf{R}$$, then... JEE Main 2024 (Online) 29th January Evening Shift Let$$f(x)=\sqrt{\lim _\limits{r \rightarrow x}\left\{\frac{2 r^2\left[(f(r))^2-f(x) f(r)\right]}{r^2-x^2}-r^3 e^{\frac{f(r)}{r}}\right\}}$$be differ... JEE Main 2023 (Online) 12th April Morning Shift Let$$[x]$$be the greatest integer$$\leq x$$. Then the number of points in the interval$$(-2,1)$$, where the function$$f(x)=|[x]|+\sqrt{x-[x]}$$i... JEE Main 2023 (Online) 10th April Morning Shift Let$$f:( - 2,2) \to R$$be defined by$$f(x) = \left\{ {\matrix{ {x[x],} & { - 2 ... JEE Main 2023 (Online) 8th April Evening Shift Let $$\mathrm{k}$$ and $$\mathrm{m}$$ be positive real numbers such that the function $$f(x)=\left\{\begin{array}{cc}3 x^{2}+k \sqrt{x+1}, & 0 0$$. T... JEE Main 2023 (Online) 6th April Morning Shift Let $$a \in \mathbb{Z}$$ and $$[\mathrm{t}]$$ be the greatest integer $$\leq \mathrm{t}$$. Then the number of points, where the function $$f(x)=[a+13 ... JEE Main 2022 (Online) 29th July Evening Shift If$$[t]$$denotes the greatest integer$$\leq t$$, then the number of points, at which the function$$f(x)=4|2 x+3|+9\left[x+\frac{1}{2}\right]-12[x+... JEE Main 2022 (Online) 28th July Morning Shift Let $$f:[0,1] \rightarrow \mathbf{R}$$ be a twice differentiable function in $$(0,1)$$ such that $$f(0)=3$$ and $$f(1)=5$$. If the line $$y=2 x+3$$ in... JEE Main 2022 (Online) 28th July Morning Shift $$\lim\limits_{x \rightarrow 0}\left(\frac{(x+2 \cos x)^{3}+2(x+2 \cos x)^{2}+3 \sin (x+2 \cos x)}{(x+2)^{3}+2(x+2)^{2}+3 \sin (x+2)}\right)^{\frac{10... JEE Main 2022 (Online) 25th July Morning Shift Let$$f(x)=\left\{\begin{array}{l}\left|4 x^{2}-8 x+5\right|, \text { if } 8 x^{2}-6 x+1 \geqslant 0 \\ {\left[4 x^{2}-8 x+5\right], \text { if } 8 x^... JEE Main 2022 (Online) 30th June Morning Shift Suppose $$\mathop {\lim }\limits_{x \to 0} {{F(x)} \over {{x^3}}}$$ exists and is equal to L, where $$F(x) = \left| {\matrix{ {a + \sin {x \over 2}... JEE Main 2022 (Online) 28th June Evening Shift If$$\mathop {\lim }\limits_{x \to 1} {{\sin (3{x^2} - 4x + 1) - {x^2} + 1} \over {2{x^3} - 7{x^2} + ax + b}} = - 2$$, then the value of (a$$-$$b) ... JEE Main 2022 (Online) 27th June Evening Shift Let [t] denote the greatest integer$$\le$$t and {t} denote the fractional part of t. The integral value of$$\alpha$$for which the left hand limit ... JEE Main 2022 (Online) 25th June Evening Shift Let$$f(x) = \left[ {2{x^2} + 1} \right]$$and$$g(x) = \left\{ {\matrix{ {2x - 3,} & {x ... JEE Main 2022 (Online) 24th June Morning Shift The number of points where the function $$f(x) = \left\{ {\matrix{ {|2{x^2} - 3x - 7|} & {if} & {x \le - 1} \cr {[4{x^2} - 1]} & {if} & { - 1... JEE Main 2021 (Online) 1st September Evening Shift Let$$f(x) = {x^6} + 2{x^4} + {x^3} + 2x + 3$$, x$$\in$$R. Then the natural number n for which$$\mathop {\lim }\limits_{x \to 1} {{{x^n}f(1) - f(x)... JEE Main 2021 (Online) 1st September Evening Shift Let [t] denote the greatest integer $$\le$$ t. The number of points where the function $$f(x) = [x]\left| {{x^2} - 1} \right| + \sin \left( {{\pi \ov... JEE Main 2021 (Online) 26th August Morning Shift Let a, b$$\in$$R, b$$\in$$0, Define a function$$f(x) = \left\{ {\matrix{ {a\sin {\pi \over 2}(x - 1),} & {for\,x \le 0} \cr {{{\tan ... JEE Main 2021 (Online) 27th July Morning Shift Let $$f:[0,3] \to R$$ be defined by $$f(x) = \min \{ x - [x],1 + [x] - x\}$$ where [x] is the greatest integer less than or equal to x. Let P denote ... JEE Main 2021 (Online) 25th July Evening Shift Consider the functionwhere P(x) is a polynomial such that P'' (x) is always a constant and P(3) = 9. If f(x) is continuous at x = 2, then P(5) is equa... JEE Main 2021 (Online) 22th July Evening Shift Let f : R $$\to$$ R be a function defined as $$f(x) = \left\{ {\matrix{ {3\left( {1 - {{|x|} \over 2}} \right)} & {if} & {|x|\, \le 2} \cr... JEE Main 2021 (Online) 20th July Evening Shift Let a function g : [ 0, 4 ]$$\to$$R be defined as$$g(x) = \left\{ {\matrix{ {\mathop {\max }\limits_{0 \le t \le x} \{ {t^3} - 6{t^2} + 9t - 3),... JEE Main 2021 (Online) 20th July Evening Shift If $$\mathop {\lim }\limits_{x \to 0} {{\alpha x{e^x} - \beta {{\log }_e}(1 + x) + \gamma {x^2}{e^{ - x}}} \over {x{{\sin }^2}x}} = 10,\alpha ,\beta ,... JEE Main 2021 (Online) 20th July Morning Shift If the value of$$\mathop {\lim }\limits_{x \to 0} {(2 - \cos x\sqrt {\cos 2x} )^{\left( {{{x + 2} \over {{x^2}}}} \right)}}$$is equal to ea, then a ... JEE Main 2021 (Online) 18th March Evening Shift Let f : R$$ \to $$R satisfy the equation f(x + y) = f(x) . f(y) for all x, y$$\in$$R and f(x)$$\ne$$0 for any x$$\in$$R. If the function f is dif... JEE Main 2021 (Online) 17th March Morning Shift If the function$$f(x) = {{\cos (\sin x) - \cos x} \over {{x^4}}}$$is continuous at each point in its domain and$$f(0) = {1 \over k}$$, then k is __... JEE Main 2021 (Online) 16th March Evening Shift Let f : R$$ \to $$R and g : R$$ \to $$R be defined as$$f(x) = \left\{ {\matrix{ {x + a,} & {x < 0} \cr {|x - 1|,} & {x \ge 0} ... JEE Main 2021 (Online) 16th March Morning Shift If $$\mathop {\lim }\limits_{x \to 0} {{a{e^x} - b\cos x + c{e^{ - x}}} \over {x\sin x}} = 2$$, then a + b + c is equal to ____________. JEE Main 2021 (Online) 25th February Evening Shift A function f is defined on [$$-$$3, 3] as$$f(x) = \left\{ {\matrix{ {\min \{ |x|,2 - {x^2}\} ,} & { - 2 \le x \le 2} \cr {[|x|],} & {2... JEE Main 2021 (Online) 25th February Evening Shift If$$\mathop {\lim }\limits_{x \to 0} {{ax - ({e^{4x}} - 1)} \over {ax({e^{4x}} - 1)}}$$exists and is equal to b, then the value of a$$-$$2b is ___... JEE Main 2021 (Online) 25th February Morning Shift The number of points, at which the function f(x) = | 2x + 1 |$$-$$3| x + 2 | + | x2 + x$$-$$2 |, x$$\in$$R is not differentiable, is __________.... JEE Main 2021 (Online) 24th February Morning Shift$$\mathop {\lim }\limits_{n \to \infty } \tan \left\{ {\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {{1 \over {1 + r + {r^2}}}} \right)} } \right\}$$... JEE Main 2020 (Online) 6th September Morning Slot Let f : R$$ \to $$R be defined as$$f\left( x \right) = \left\{ {\matrix{ {{x^5}\sin \left( {{1 \over x}} \right) + 5{x^2},} & {x < 0} \c... JEE Main 2020 (Online) 5th September Morning Slot Let $$f(x) = x.\left[ {{x \over 2}} \right]$$, for -10< x < 10, where [t] denotes the greatest integer function. Then the number of points of d... JEE Main 2020 (Online) 4th September Morning Slot Suppose a differentiable function f(x) satisfies the identity f(x+y) = f(x) + f(y) + xy2 + x2y, for all real x and y. $$\mathop {\lim }\limits_{x \to ... JEE Main 2020 (Online) 3rd September Morning Slot If$$\mathop {\lim }\limits_{x \to 0} \left\{ {{1 \over {{x^8}}}\left( {1 - \cos {{{x^2}} \over 2} - \cos {{{x^2}} \over 4} + \cos {{{x^2}} \over 2}\c... JEE Main 2020 (Online) 2nd September Morning Slot If $$\mathop {\lim }\limits_{x \to 1} {{x + {x^2} + {x^3} + ... + {x^n} - n} \over {x - 1}}$$ = 820, (n $$\in$$ N) then the value of n is equal to _... JEE Main 2020 (Online) 7th January Evening Slot If the function ƒ defined on $$\left( { - {1 \over 3},{1 \over 3}} \right)$$ by f(x) = $$\left\{ {\matrix{ {{1 \over x}{{\log }_e}\left( {{{1 + 3x}... JEE Main 2020 (Online) 7th January Morning Slot Let S be the set of points where the function, ƒ(x) = |2-|x-3||, x$$ \in $$R is not differentiable. Then$$\sum\limits_{x \in S} {f(f(x))} $$is eq... JEE Main 2020 (Online) 7th January Morning Slot$$\mathop {\lim }\limits_{x \to 2} {{{3^x} + {3^{3 - x}} - 12} \over {{3^{ - x/2}} - {3^{1 - x}}}}$$is equal to_______. ## MCQ (Single Correct Answer) JEE Main 2024 (Online) 9th April Evening Shift$$\lim _\limits{x \rightarrow 0} \frac{e-(1+2 x)^{\frac{1}{2 x}}}{x}$$is equal to JEE Main 2024 (Online) 1st February Evening Shift Let f(x)=\left|2 x^2+5\right| x|-3|, x \in \mathbf{R}. If \mathrm{m} and \mathrm{n} denote the number of points where f is not continuous and ... JEE Main 2024 (Online) 1st February Evening Shift Let f(x)=\left\{\begin{array}{l}x-1, x \text { is even, } \\ 2 x, \quad x \text { is odd, }\end{array} x \in \mathbf{N}\right.. If for some \mathr... JEE Main 2024 (Online) 1st February Morning Shift Let f: \mathbf{R} \rightarrow \mathbf{R} be defined as :$$ f(x)= \begin{cases}\frac{a-b \cos 2 x}{x^2} ; & x1\end{cases} $$If f is continuous e... JEE Main 2024 (Online) 31st January Evening Shift Consider the function$$f:(0, \infty) \rightarrow \mathbb{R}$$defined by$$f(x)=e^{-\left|\log _e x\right|}$$. If$$m$$and$$n$$be respectively the... JEE Main 2024 (Online) 31st January Morning Shift$$\lim _\limits{x \rightarrow 0} \frac{e^{2|\sin x|}-2|\sin x|-1}{x^2}$$JEE Main 2024 (Online) 31st January Morning Shift Let$$g(x)$$be a linear function and$$f(x)=\left\{\begin{array}{cl}g(x) & , x \leq 0 \\ \left(\frac{1+x}{2+x}\right)^{\frac{1}{x}} & , x>0\end{array... JEE Main 2024 (Online) 27th January Evening Shift Consider the function $$f:(0,2) \rightarrow \mathbf{R}$$ defined by $$f(x)=\frac{x}{2}+\frac{2}{x}$$ and the function $$g(x)$$ defined by $$g(x)=\left... JEE Main 2024 (Online) 27th January Evening Shift$$\text { If } \lim _\limits{x \rightarrow 0} \frac{3+\alpha \sin x+\beta \cos x+\log _e(1-x)}{3 \tan ^2 x}=\frac{1}{3} \text {, then } 2 \alpha-\beta... JEE Main 2024 (Online) 27th January Morning Shift Consider the function. $$f(x)=\left\{\begin{array}{cc} \frac{\mathrm{a}\left(7 x-12-x^2\right)}{\mathrm{b}\left|x^2-7 x+12\right|} & , x3 \\\\ \mathr... JEE Main 2024 (Online) 27th January Morning Shift If \mathrm{a}=\lim\limits_{x \rightarrow 0} \frac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2}}{x^4} and \mathrm{b}=\lim\limits _{x \rightarrow 0} \frac{\sin ^2 ... JEE Main 2023 (Online) 15th April Morning Shift Let [x] denote the greatest integer function and f(x)=\max \{1+x+[x], 2+x, x+2[x]\}, 0 \leq x \leq 2. Let m be the number of points in [0,2], ... JEE Main 2023 (Online) 13th April Evening Shift If$$\lim_\limits{x \rightarrow 0} \frac{e^{a x}-\cos (b x)-\frac{cx e^{-c x}}{2}}{1-\cos (2 x)}=17$$, then$$5 a^{2}+b^{2}$$is equal to JEE Main 2023 (Online) 11th April Evening Shift Let$$f$$and$$g$$be two functions defined by$$f(x)=\left\{\begin{array}{cc}x+1, & x Then $$(g \circ f)(x)$$ is :... JEE Main 2023 (Online) 11th April Morning Shift Let $$f(x)=\left[x^{2}-x\right]+|-x+[x]|$$, where $$x \in \mathbb{R}$$ and $$[t]$$ denotes the greatest integer less than or equal to $$t$$. Then, $$f... JEE Main 2023 (Online) 8th April Evening Shift If$$\alpha > \beta > 0$$are the roots of the equation$$a x^{2}+b x+1=0$$, and$$\lim_\limits{x \rightarrow \frac{1}{\alpha}}\left(\frac{1-\cos \lef... JEE Main 2023 (Online) 8th April Morning Shift $$\lim_\limits{x \rightarrow 0}\left(\left(\frac{\left(1-\cos ^{2}(3 x)\right.}{\cos ^{3}(4 x)}\right)\left(\frac{\sin ^{3}(4 x)}{\left(\log _{e}(2 x+... JEE Main 2023 (Online) 6th April Morning Shift Let$$a_{1}, a_{2}, a_{3}, \ldots, a_{\mathrm{n}}$$be$$\mathrm{n}$$positive consecutive terms of an arithmetic progression. If$$\mathrm{d} > 0$$i... JEE Main 2023 (Online) 31st January Evening Shift$$ \lim\limits_{x \rightarrow \infty} \frac{(\sqrt{3 x+1}+\sqrt{3 x-1})^6+(\sqrt{3 x+1}-\sqrt{3 x-1})^6}{\left(x+\sqrt{x^2-1}\right)^6+\left(x-\sqrt{x... JEE Main 2023 (Online) 30th January Evening Shift Let$f, g$and$h$be the real valued functions defined on$\mathbb{R}$as$f(x)=\left\{\begin{array}{cc}\frac{x}{|x|}, & x \neq 0 \\ 1, & x=0\end{arr...
JEE Main 2023 (Online) 30th January Morning Shift
Suppose $$f: \mathbb{R} \rightarrow(0, \infty)$$ be a differentiable function such that $$5 f(x+y)=f(x) \cdot f(y), \forall x, y \in \mathbb{R}$$. If ...
JEE Main 2023 (Online) 29th January Morning Shift
Let $$x=2$$ be a root of the equation $$x^2+px+q=0$$ and $$f(x) = \left\{ {\matrix{ {{{1 - \cos ({x^2} - 4px + {q^2} + 8q + 16)} \over {{{(x - 2p)}... JEE Main 2023 (Online) 25th January Evening Shift If the function$$f(x) = \left\{ {\matrix{ {(1 + |\cos x|)^{\lambda \over {|\cos x|}}} & , & {0 is continuous at $$x = {\pi \over 2}$$, then $$9... JEE Main 2023 (Online) 25th January Morning Shift The value of$$\mathop {\lim }\limits_{n \to \infty } {{1 + 2 - 3 + 4 + 5 - 6\, + \,.....\, + \,(3n - 2) + (3n - 1) - 3n} \over {\sqrt {2{n^4} + 4n + ...
JEE Main 2023 (Online) 24th January Evening Shift
The set of all values of $$a$$ for which $$\mathop {\lim }\limits_{x \to a} ([x - 5] - [2x + 2]) = 0$$, where [$$\alpha$$] denotes the greatest intege...
JEE Main 2023 (Online) 24th January Morning Shift
$$\mathop {\lim }\limits_{t \to 0} {\left( {{1^{{1 \over {{{\sin }^2}t}}}} + {2^{{1 \over {{{\sin }^2}t}}}}\, + \,...\, + \,{n^{{1 \over {{{\sin }^2}t... JEE Main 2023 (Online) 24th January Morning Shift Let$$f(x) = \left\{ {\matrix{ {{x^2}\sin \left( {{1 \over x}} \right)} & {,\,x \ne 0} \cr 0 & {,\,x = 0} \cr } } \right.$$Then at$$x=0$... JEE Main 2022 (Online) 29th July Evening Shift $$\text { Let the function } f(x)=\left\{\begin{array}{cl} \frac{\log _{e}(1+5 x)-\log _{e}(1+\alpha x)}{x} & ;\text { if } x \neq 0 \\ 10 & ; \text ... JEE Main 2022 (Online) 29th July Morning Shift If$$\lim\limits_{x \rightarrow 0} \frac{\alpha \mathrm{e}^{x}+\beta \mathrm{e}^{-x}+\gamma \sin x}{x \sin ^{2} x}=\frac{2}{3}$$, where$$\alpha, \bet... JEE Main 2022 (Online) 29th July Morning Shift The number of points, where the function $$f: \mathbf{R} \rightarrow \mathbf{R}$$, $$f(x)=|x-1| \cos |x-2| \sin |x-1|+(x-3)\left|x^{2}-5 x+4\right|$$,... JEE Main 2022 (Online) 28th July Evening Shift The function $$f: \mathbb{R} \rightarrow \mathbb{R}$$ defined by $$f(x)=\lim\limits_{n \rightarrow \infty} \frac{\cos (2 \pi x)-x^{2 n} \sin (x-1)}{1+... JEE Main 2022 (Online) 27th July Evening Shift If for$$\mathrm{p} \neq \mathrm{q} \neq 0$$, the function$$f(x)=\frac{\sqrt[7]{\mathrm{p}(729+x)}-3}{\sqrt[3]{729+\mathrm{q} x}-9}$$is continuous a... JEE Main 2022 (Online) 26th July Evening Shift Let$$\beta=\mathop {\lim }\limits_{x \to 0} \frac{\alpha x-\left(e^{3 x}-1\right)}{\alpha x\left(e^{3 x}-1\right)}$$for some$$\alpha \in \mathbb{R}... JEE Main 2022 (Online) 26th July Morning Shift Let f : R $$\to$$ R be a continuous function such that $$f(3x) - f(x) = x$$. If $$f(8) = 7$$, then $$f(14)$$ is equal to : JEE Main 2022 (Online) 26th July Morning Shift If the function $$f(x) = \left\{ {\matrix{ {{{{{\log }_e}(1 - x + {x^2}) + {{\log }_e}(1 + x + {x^2})} \over {\sec x - \cos x}}} & , & {x \in \left... JEE Main 2022 (Online) 26th July Morning Shift If$$f(x) = \left\{ {\matrix{ {x + a} & , & {x \le 0} \cr {|x - 4|} & , & {x > 0} \cr } } \right.$$and$$g(x) = \left\{ {\matrix{ {x +... JEE Main 2022 (Online) 26th July Morning Shift Let $$f(x) = \left\{ {\matrix{ {{x^3} - {x^2} + 10x - 7,} & {x \le 1} \cr { - 2x + {{\log }_2}({b^2} - 4),} & {x > 1} \cr } } \right.$$. T... JEE Main 2022 (Online) 25th July Evening Shift $$\lim\limits_{x \rightarrow \frac{\pi}{4}} \frac{8 \sqrt{2}-(\cos x+\sin x)^{7}}{\sqrt{2}-\sqrt{2} \sin 2 x}$$ is equal to JEE Main 2022 (Online) 25th July Morning Shift If $$\mathop {\lim }\limits_{n \to \infty } \left( {\sqrt {{n^2} - n - 1} + n\alpha + \beta } \right) = 0$$, then $$8(\alpha+\beta)$$ is equal to : ... JEE Main 2022 (Online) 29th June Evening Shift The value of $$\mathop {\lim }\limits_{x \to 1} {{({x^2} - 1){{\sin }^2}(\pi x)} \over {{x^4} - 2{x^3} + 2x - 1}}$$ is equal to: JEE Main 2022 (Online) 28th June Evening Shift Let f, g : R $$\to$$ R be functions defined by $$f(x) = \left\{ {\matrix{ {[x]} & , & {x ... JEE Main 2022 (Online) 28th June Evening Shift The value of$$\mathop {\lim }\limits_{n \to \infty } 6\tan \left\{ {\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {{1 \over {{r^2} + 3r + 3}}} \right... JEE Main 2022 (Online) 28th June Morning Shift Let f : R $$\to$$ R be defined as $$f(x) = \left[ {\matrix{ {[{e^x}],} & {x where a, b, c$$\in$$R and [t] denotes greatest integer less than or ... JEE Main 2022 (Online) 27th June Morning Shift Let a be an integer such that$$\mathop {\lim }\limits_{x \to 7} {{18 - [1 - x]} \over {[x - 3a]}}$$exists, where [t] is greatest integer$$\le$$t. ... JEE Main 2022 (Online) 26th June Evening Shift$$\mathop {\lim }\limits_{x \to 0} {{\cos (\sin x) - \cos x} \over {{x^4}}}$$is equal to : JEE Main 2022 (Online) 26th June Evening Shift Let f(x) = min {1, 1 + x sin x}, 0$$\le$$x$$\le$$2$$\pi $$. If m is the number of points, where f is not differentiable and n is the number of poi... JEE Main 2022 (Online) 26th June Morning Shift$$\mathop {\lim }\limits_{x \to {1 \over {\sqrt 2 }}} {{\sin ({{\cos }^{ - 1}}x) - x} \over {1 - \tan ({{\cos }^{ - 1}}x)}}$$is equal to : JEE Main 2022 (Online) 26th June Morning Shift Let f, g : R$$\to$$R be two real valued functions defined as$$f(x) = \left\{ {\matrix{ { - |x + 3|} & , & {x 1 and k2 are real constants. If (go... JEE Main 2022 (Online) 25th June Evening Shift $$\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{{\tan }^2}x\left( {{{(2{{\sin }^2}x + 3\sin x + 4)}^{{1 \over 2}}} - {{({{\sin }^2}x + 6\sin ... JEE Main 2022 (Online) 25th June Morning Shift Let f(x) be a polynomial function such that$$f(x) + f'(x) + f''(x) = {x^5} + 64$$. Then, the value of$$\mathop {\lim }\limits_{x \to 1} {{f(x)} \ove... JEE Main 2022 (Online) 24th June Evening Shift Let $$f(x) = \left\{ {\matrix{ {{{\sin (x - [x])} \over {x - [x]}}} & {,\,x \in ( - 2, - 1)} \cr {\max \{ 2x,3[|x|]\} } & {,\,|x| where [t] d... JEE Main 2021 (Online) 31st August Evening Shift If$$\alpha = \mathop {\lim }\limits_{x \to {\pi \over 4}} {{{{\tan }^3}x - \tan x} \over {\cos \left( {x + {\pi \over 4}} \right)}}$$and$$\beta ... JEE Main 2021 (Online) 31st August Evening Shift Let f be any continuous function on [0, 2] and twice differentiable on (0, 2). If f(0) = 0, f(1) = 1 and f(2) = 2, then JEE Main 2021 (Online) 31st August Morning Shift The function $$f(x) = \left| {{x^2} - 2x - 3} \right|\,.\,{e^{\left| {9{x^2} - 12x + 4} \right|}}$$ is not differentiable at exactly : JEE Main 2021 (Online) 31st August Morning Shift If the function $$f(x) = \left\{ {\matrix{ {{1 \over x}{{\log }_e}\left( {{{1 + {x \over a}} \over {1 - {x \over b}}}} \right)} & , & {x &l... JEE Main 2021 (Online) 31st August Morning Shift$$\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}\left( {\pi {{\cos }^4}x} \right)} \over {{x^4}}}$$is equal to : JEE Main 2021 (Online) 27th August Evening Shift If$$\mathop {\lim }\limits_{x \to \infty } \left( {\sqrt {{x^2} - x + 1} - ax} \right) = b$$, then the ordered pair (a, b) is : JEE Main 2021 (Online) 27th August Morning Shift If$$\alpha$$,$$\beta$$are the distinct roots of x2 + bx + c = 0, then$$\mathop {\lim }\limits_{x \to \beta } {{{e^{2({x^2} + bx + c)}} - 1 - 2({x^... JEE Main 2021 (Online) 26th August Evening Shift Let [t] denote the greatest integer less than or equal to t. Let f(x) = x $$-$$ [x], g(x) = 1 $$-$$ x + [x], and h(x) = min{f(x), g(x)}, x $$\in$$ [$$... JEE Main 2021 (Online) 26th August Evening Shift$$\mathop {\lim }\limits_{x \to 2} \left( {\sum\limits_{n = 1}^9 {{x \over {n(n + 1){x^2} + 2(2n + 1)x + 4}}} } \right)$$is equal to : JEE Main 2021 (Online) 27th July Evening Shift The value of$$\mathop {\lim }\limits_{x \to 0} \left( {{x \over {\root 8 \of {1 - \sin x} - \root 8 \of {1 + \sin x} }}} \right)$$is equal to :... JEE Main 2021 (Online) 27th July Evening Shift Let$$f:[0,\infty ) \to [0,3]$$be a function defined by$$f(x) = \left\{ {\matrix{ {\max \{ \sin t:0 \le t \le x\} ,} & {0 \le x \le \pi } \c... JEE Main 2021 (Online) 27th July Morning Shift Let $$f:\left( { - {\pi \over 4},{\pi \over 4}} \right) \to R$$ be defined as $$f(x) = \left\{ {\matrix{ {{{(1 + |\sin x|)}^{{{3a} \over {|\sin x... JEE Main 2021 (Online) 27th July Morning Shift Let f : R$$\to$$R be a function such that f(2) = 4 and f'(2) = 1. Then, the value of$$\mathop {\lim }\limits_{x \to 2} {{{x^2}f(2) - 4f(x)} \over {... JEE Main 2021 (Online) 25th July Morning Shift Let f : R $$\to$$ R be defined as$$f(x) = \left\{ {\matrix{ {{{\lambda \left| {{x^2} - 5x + 6} \right|} \over {\mu (5x - {x^2} - 6)}},} & {x &l... JEE Main 2021 (Online) 22th July Evening Shift Let f : R$$\to$$R be defined as$$f(x) = \left\{ {\matrix{ {{{{x^3}} \over {{{(1 - \cos 2x)}^2}}}{{\log }_e}\left( {{{1 + 2x{e^{ - 2x}}} \over {{... JEE Main 2021 (Online) 20th July Evening Shift If $$f:R \to R$$ is given by $$f(x) = x + 1$$, then the value of $$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\left[ {f(0) + f\left( {{5 \over ... JEE Main 2021 (Online) 20th July Morning Shift Let a function f : R$$\to$$R be defined as$$f(x) = \left\{ {\matrix{ {\sin x - {e^x}} & {if} & {x \le 0} \cr {a + [ - x]} & {if... JEE Main 2021 (Online) 18th March Evening Shift Let f : R $$\to$$ R be a function defined as$$f(x) = \left\{ \matrix{ {{\sin (a + 1)x + \sin 2x} \over {2x}},if\,x < 0 \hfill \cr b,\,if\,x\... JEE Main 2021 (Online) 18th March Morning Shift If$$\mathop {\lim }\limits_{x \to 0} {{{{\sin }^{ - 1}}x - {{\tan }^{ - 1}}x} \over {3{x^3}}}$$is equal to L, then the value of (6L + 1) is JEE Main 2021 (Online) 18th March Morning Shift If$$f(x) = \left\{ {\matrix{ {{1 \over {|x|}}} & {;\,|x|\, \ge 1} \cr {a{x^2} + b} & {;\,|x|\, < 1} \cr } } \right.$$is diffe... JEE Main 2021 (Online) 17th March Evening Shift The value of the limit$$\mathop {\lim }\limits_{\theta \to 0} {{\tan (\pi {{\cos }^2}\theta )} \over {\sin (2\pi {{\sin }^2}\theta )}}$$is equal to... JEE Main 2021 (Online) 17th March Evening Shift The value of$$\mathop {\lim }\limits_{n \to \infty } {{[r] + [2r] + ... + [nr]} \over {{n^2}}}$$, where r is a non-zero real number and [r] denotes t... JEE Main 2021 (Online) 17th March Morning Shift The value of$$\mathop {\lim }\limits_{x \to {0^ + }} {{{{\cos }^{ - 1}}(x - {{[x]}^2}).{{\sin }^{ - 1}}(x - {{[x]}^2})} \over {x - {x^3}}}$$, where [... JEE Main 2021 (Online) 16th March Evening Shift Let f : S$$ \to $$S where S = (0,$$\infty $$) be a twice differentiable function such that f(x + 1) = xf(x). If g : S$$ \to $$R be defined as g(x... JEE Main 2021 (Online) 16th March Evening Shift Let$$\alpha\in$$R be such that the function$$f(x) = \left\{ {\matrix{ {{{{{\cos }^{ - 1}}(1 - {{\{ x\} }^2}){{\sin }^{ - 1}}(1 - \{ x\} )} ... JEE Main 2021 (Online) 16th March Morning Shift Let $${S_k} = \sum\limits_{r = 1}^k {{{\tan }^{ - 1}}\left( {{{{6^r}} \over {{2^{2r + 1}} + {3^{2r + 1}}}}} \right)}$$. Then $$\mathop {\lim }\limits... JEE Main 2021 (Online) 16th March Morning Shift Let the functions f : R$$ \to $$R and g : R$$ \to $$R be defined as :$$f(x) = \left\{ {\matrix{ {x + 2,} & {x < 0} \cr {{x^2},} &am... JEE Main 2021 (Online) 26th February Evening Shift Let f(x) be a differentiable function at x = a with f'(a) = 2 and f(a) = 4. Then $$\mathop {\lim }\limits_{x \to a} {{xf(a) - af(x)} \over {x - a}}$$ ... JEE Main 2021 (Online) 26th February Evening Shift Let $$f(x) = {\sin ^{ - 1}}x$$ and $$g(x) = {{{x^2} - x - 2} \over {2{x^2} - x - 6}}$$. If $$g(2) = \mathop {\lim }\limits_{x \to 2} g(x)$$, then the ... JEE Main 2021 (Online) 26th February Evening Shift Let f : R $$\to$$ R be defined as $$f(x) = \left\{ \matrix{ 2\sin \left( { - {{\pi x} \over 2}} \right),if\,x < - 1 \hfill \cr |a{x^2} + x ... JEE Main 2021 (Online) 26th February Morning Shift The value of$$\mathop {\lim }\limits_{h \to 0} 2\left\{ {{{\sqrt 3 \sin \left( {{\pi \over 6} + h} \right) - \cos \left( {{\pi \over 6} + h} \right... JEE Main 2021 (Online) 25th February Morning Shift $$\mathop {\lim }\limits_{n \to \infty } {\left( {1 + {{1 + {1 \over 2} + ........ + {1 \over n}} \over {{n^2}}}} \right)^n}$$ is equal to : JEE Main 2021 (Online) 24th February Morning Shift If f : R $$\to$$ R is a function defined by f(x)= [x - 1] $$\cos \left( {{{2x - 1} \over 2}} \right)\pi$$, where [.] denotes the greatest integer f... JEE Main 2020 (Online) 6th September Evening Slot Let f : R $$\to$$ R be a function defined by f(x) = max {x, x2}. Let S denote the set of all points in R, where f is not differentiable. Then :... JEE Main 2020 (Online) 6th September Evening Slot For all twice differentiable functions f : R $$\to$$ R, with f(0) = f(1) = f'(0) = 0 JEE Main 2020 (Online) 5th September Evening Slot $$\mathop {\lim }\limits_{x \to 0} {{x\left( {{e^{\left( {\sqrt {1 + {x^2} + {x^4}} - 1} \right)/x}} - 1} \right)} \over {\sqrt {1 + {x^2} + {x^4}} ... JEE Main 2020 (Online) 5th September Morning Slot If the function$$f\left( x \right) = \left\{ {\matrix{ {{k_1}{{\left( {x - \pi } \right)}^2} - 1,} & {x \le \pi } \cr {{k_2}\cos x,} &... JEE Main 2020 (Online) 5th September Morning Slot If $$\alpha$$ is positive root of the equation, p(x) = x2 - x - 2 = 0, then $$\mathop {\lim }\limits_{x \to {\alpha ^ + }} {{\sqrt {1 - \cos \left( {... JEE Main 2020 (Online) 4th September Evening Slot Let$$f:\left( {0,\infty } \right) \to \left( {0,\infty } \right)$$be a differentiable function such that f(1) = e and$$\mathop {\lim }\limits_{t \t... JEE Main 2020 (Online) 4th September Evening Slot The function $$f(x) = \left\{ {\matrix{ {{\pi \over 4} + {{\tan }^{ - 1}}x,} & {\left| x \right| \le 1} \cr {{1 \over 2}\left( {\left| x ... JEE Main 2020 (Online) 3rd September Evening Slot$$\mathop {\lim }\limits_{x \to a} {{{{\left( {a + 2x} \right)}^{{1 \over 3}}} - {{\left( {3x} \right)}^{{1 \over 3}}}} \over {{{\left( {3a + x} \righ... JEE Main 2020 (Online) 3rd September Morning Slot Let [t] denote the greatest integer $$\le$$ t. If for some $$\lambda$$ $$\in$$ R - {1, 0}, $$\mathop {\lim }\limits_{x \to 0} \left| {{{1 - x + \... JEE Main 2020 (Online) 2nd September Evening Slot$$\mathop {\lim }\limits_{x \to 0} {\left( {\tan \left( {{\pi \over 4} + x} \right)} \right)^{{1 \over x}}}$$is equal to : JEE Main 2020 (Online) 2nd September Morning Slot If a function f(x) defined by$$f\left( x \right) = \left\{ {\matrix{ {a{e^x} + b{e^{ - x}},} & { - 1 \le x < 1} \cr {c{x^2},} & {1... JEE Main 2020 (Online) 9th January Evening Slot Let [t] denote the greatest integer $$\le$$ t and $$\mathop {\lim }\limits_{x \to 0} x\left[ {{4 \over x}} \right] = A$$. Then the function, f(x) = ... JEE Main 2020 (Online) 9th January Morning Slot Let ƒ be any function continuous on [a, b] and twice differentiable on (a, b). If for all x $$\in$$ (a, b), ƒ'(x) > 0 and ƒ''(x) < 0, then for... JEE Main 2020 (Online) 9th January Morning Slot If $$f(x) = \left\{ {\matrix{ {{{\sin (a + 2)x + \sin x} \over x};} & {x < 0} \cr {b\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\... JEE Main 2020 (Online) 8th January Evening Slot Let S be the set of all functions ƒ : [0,1]$$ \to $$R, which are continuous on [0,1] and differentiable on (0,1). Then for every ƒ in S, there exist... JEE Main 2020 (Online) 8th January Morning Slot$$\mathop {\lim }\limits_{x \to 0} {\left( {{{3{x^2} + 2} \over {7{x^2} + 2}}} \right)^{{1 \over {{x^2}}}}}$$is equal to JEE Main 2019 (Online) 12th April Evening Slot$$\mathop {\lim }\limits_{x \to 0} {{x + 2\sin x} \over {\sqrt {{x^2} + 2\sin x + 1} - \sqrt {{{\sin }^2}x - x + 1} }}$$is : JEE Main 2019 (Online) 12th April Evening Slot Let f(x) = 5 – |x – 2| and g(x) = |x + 1|, x$$ \in $$R. If f(x) attains maximum value at$$\alpha $$and g(x) attains minimum value at$$\beta $$, t... JEE Main 2019 (Online) 12th April Morning Slot If$$\alpha $$and$$\beta $$are the roots of the equation 375x2 – 25x – 2 = 0, then$$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {... JEE Main 2019 (Online) 10th April Evening Slot If $$\mathop {\lim }\limits_{x \to 1} {{{x^2} - ax + b} \over {x - 1}} = 5$$, then a + b is equal to : JEE Main 2019 (Online) 10th April Morning Slot If $$\mathop {\lim }\limits_{x \to 1} {{{x^4} - 1} \over {x - 1}} = \mathop {\lim }\limits_{x \to k} {{{x^3} - {k^3}} \over {{x^2} - {k^2}}}$$, then ... JEE Main 2019 (Online) 10th April Morning Slot Let f : R $$\to$$ R be differentiable at c $$\in$$ R and f(c) = 0. If g(x) = |f(x)| , then at x = c, g is : JEE Main 2019 (Online) 10th April Morning Slot If$$f(x) = \left\{ {\matrix{ {{{\sin (p + 1)x + \sin x} \over x}} & {,x < 0} \cr q & {,x = 0} \cr {{{\sqrt {x + {x^2}} - \sqr... JEE Main 2019 (Online) 9th April Evening Slot If$$f(x) = [x] - \left[ {{x \over 4}} \right]$$,x$$ \in $$4 , where [x] denotes the greatest integer function, then JEE Main 2019 (Online) 9th April Evening Slot If the function$$f(x) = \left\{ {\matrix{ {a|\pi - x| + 1,x \le 5} \cr {b|x - \pi | + 3,x > 5} \cr } } \right.$$is continuous at x =... JEE Main 2019 (Online) 9th April Morning Slot Let ƒ(x) = 15 – |x – 10|; x$$ \in $$R. Then the set of all values of x, at which the function, g(x) = ƒ(ƒ(x)) is not differentiable, is : JEE Main 2019 (Online) 9th April Morning Slot If the function ƒ defined on ,$$\left( {{\pi \over 6},{\pi \over 3}} \right)$$by$$$f(x) = \left\{ {\matrix{ {{{\sqrt 2 {\mathop{\rm cosx}\noli...
JEE Main 2019 (Online) 8th April Evening Slot
Let ƒ : R $$\to$$ R be a differentiable function satisfying ƒ'(3) + ƒ'(2) = 0. Then $$\mathop {\lim }\limits_{x \to 0} {\left( {{{1 + f(3 + x) - f(3... JEE Main 2019 (Online) 8th April Evening Slot Let ƒ : [–1,3]$$ \to $$R be defined as$$f(x) = \left\{ {\matrix{ {\left| x \right| + \left[ x \right]} & , & { - 1 \le x < 1} \cr ...
JEE Main 2019 (Online) 8th April Morning Slot
$$\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}x} \over {\sqrt 2 - \sqrt {1 + \cos x} }}$$ equals:
JEE Main 2019 (Online) 12th January Evening Slot
$$\mathop {\lim }\limits_{x \to {1^ - }} {{\sqrt \pi - \sqrt {2{{\sin }^{ - 1}}x} } \over {\sqrt {1 - x} }}$$ is equal to :
JEE Main 2019 (Online) 12th January Evening Slot
Let f be a differentiable function such that f(1) = 2 and f '(x) = f(x) for all x $$\in$$ R R. If h(x) = f(f(x)), then h'(1) is equal to :
JEE Main 2019 (Online) 12th January Morning Slot
$$\mathop {\lim }\limits_{x \to \pi /4} {{{{\cot }^3}x - \tan x} \over {\cos \left( {x + {\pi \over 4}} \right)}}$$ is :
JEE Main 2019 (Online) 12th January Morning Slot
Let S be the set of all points in (–$$\pi$$, $$\pi$$) at which the function, f(x) = min{sin x, cos x} is not differentiable. Then S is a subset of w...
JEE Main 2019 (Online) 11th January Evening Slot
Let K be the set of all real values of x where the function f(x) = sin |x| – |x| + 2(x – $$\pi$$) cos |x| is not differentiable. Then the set K is eq...
JEE Main 2019 (Online) 11th January Evening Slot
$$\mathop {\lim }\limits_{x \to 0} {{x\cot \left( {4x} \right)} \over {{{\sin }^2}x{{\cot }^2}\left( {2x} \right)}}$$ is equal to :
JEE Main 2019 (Online) 11th January Morning Slot
Let $$f\left( x \right) = \left\{ {\matrix{ { - 1} & { - 2 \le x < 0} \cr {{x^2} - 1,} & {0 \le x \le 2} \cr } } \right.$$ and...
JEE Main 2019 (Online) 11th January Morning Slot
Let [x] denote the greatest integer less than or equal to x. Then $$\mathop {\lim }\limits_{x \to 0} {{\tan \left( {\pi {{\sin }^2}x} \right) + {{\lef... JEE Main 2019 (Online) 10th January Evening Slot Let f : ($$-$$1, 1)$$ \to $$R be a function defined by f(x) = max$$\left\{ { - \left| x \right|, - \sqrt {1 - {x^2}} } \right\}.$$If K be the set ... JEE Main 2019 (Online) 10th January Morning Slot For each t$$ \in $$R , let [t] be the greatest integer less than or equal to t Then$$\mathop {\lim }\limits_{x \to 1^ + } {{\left( {1 - ...
JEE Main 2019 (Online) 10th January Morning Slot
Let  $$f\left( x \right) = \left\{ {\matrix{ {\max \left\{ {\left| x \right|,{x^2}} \right\}} & {\left| x \right| \le 2} \cr {8 ... JEE Main 2019 (Online) 9th January Evening Slot For each x$$ \in $$R, let [x] be the greatest integer less than or equal to x. Then$$\mathop {\lim }\limits_{x \to {0^ - }} \,\,{{x\left( {\left[ x ...
JEE Main 2019 (Online) 9th January Morning Slot
Let f : R $$\to$$ R be a function defined as $$f(x) = \left\{ {\matrix{ 5 & ; & {x \le 1} \cr {a + bx} & ; & {1 < x <... JEE Main 2019 (Online) 9th January Morning Slot$$\mathop {\lim }\limits_{y \to 0} {{\sqrt {1 + \sqrt {1 + {y^4}} } - \sqrt 2 } \over {{y^4}}}$$JEE Main 2018 (Online) 16th April Morning Slot$$\mathop {\lim }\limits_{x \to 0} \,\,{{{{\left( {27 + x} \right)}^{{1 \over 3}}} - 3} \over {9 - {{\left( {27 + x} \right)}^{{2 \over 3}}}}}$$equal... JEE Main 2018 (Online) 16th April Morning Slot If the function f defined as$$f\left( x \right) = {1 \over x} - {{k - 1} \over {{e^{2x}} - 1}},x \ne 0,$$is continuous at x = 0, then the ordered ... JEE Main 2018 (Offline) For each t$$ \in R$$, let [t] be the greatest integer less than or equal to t. Then$$\mathop {\lim }\limits_{x \to {0^ + }} x\left( {\left[ {{1 \ove...
JEE Main 2018 (Offline)
Let S = { t $$\in R:f(x) = \left| {x - \pi } \right|.\left( {{e^{\left| x \right|}} - 1} \right)$$$$\sin \left| x \right|$$ is not differentiable at ...
JEE Main 2018 (Online) 15th April Evening Slot
$$\mathop {\lim }\limits_{x \to 0} {{x\tan 2x - 2x\tan x} \over {{{\left( {1 - \cos 2x} \right)}^2}}}$$ equals :
JEE Main 2018 (Online) 15th April Evening Slot
Let f(x) = $$\left\{ {\matrix{ {{{\left( {x - 1} \right)}^{{1 \over {2 - x}}}},} & {x > 1,x \ne 2} \cr {k\,\,\,\,\,\,\,\,\,\,\,\,\,\,} ... JEE Main 2018 (Online) 15th April Evening Slot Let f(x) be a polynomial of degree$$4$$having extreme values at$$x = 1$$and$$x = 2.$$If$$\mathop {lim}\limits_{x \to 0} \left( {{{f\left...
JEE Main 2018 (Online) 15th April Morning Slot
Let S = {($$\lambda$$, $$\mu$$) $$\in$$ R $$\times$$ R : f(t) = (|$$\lambda$$| e|t| $$-$$ $$\mu$$). sin (2|t|), t $$\in$$ R, is a differen...
JEE Main 2017 (Online) 9th April Morning Slot
The value of k for which the function $$f\left( x \right) = \left\{ {\matrix{ {{{\left( {{4 \over 5}} \right)}^{{{\tan \,4x} \over {\tan \,5x}}}}\... JEE Main 2017 (Online) 8th April Morning Slot$$\mathop {\lim }\limits_{x \to 3} {{\sqrt {3x} - 3} \over {\sqrt {2x - 4} - \sqrt 2 }}$$is equal to : JEE Main 2017 (Offline)$$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{\cot x - \cos x} \over {{{\left( {\pi - 2x} \right)}^3}}}$$equals JEE Main 2016 (Online) 10th April Morning Slot$$\mathop {\lim }\limits_{x \to 0} \,{{{{\left( {1 - \cos 2x} \right)}^2}} \over {2x\,\tan x\, - x\tan 2x}}$$is : JEE Main 2016 (Online) 10th April Morning Slot Let a, b$$ \in $$R, (a$$ \ne $$0). If the function f defined as$$f\left( x \right) = \left\{ {\matrix{ {{{2{x^2}} \over a}\,\,,} & {0 \le ...
JEE Main 2016 (Online) 9th April Morning Slot
If    $$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} - {4 \over {{x^2}}}} \right)^{2x}} = {e^3},$$ then 'a' is equal to :...
JEE Main 2016 (Online) 9th April Morning Slot
If the function f(x) = $$\left\{ {\matrix{ { - x} & {x < 1} \cr {a + {{\cos }^{ - 1}}\left( {x + b} \right),} & {1 \le x \le 2} \... JEE Main 2016 (Offline) For$$x \in \,R,\,\,f\left( x \right) = \left| {\log 2 - \sin x} \right|\,\,$$and$$\,\,g\left( x \right) = f\left( {f\left( x \right)} \right),\,\,...
JEE Main 2016 (Offline)
Let $$p = \mathop {\lim }\limits_{x \to {0^ + }} {\left( {1 + {{\tan }^2}\sqrt x } \right)^{{1 \over {2x}}}}$$ then $$log$$ $$p$$ is equal to :
JEE Main 2015 (Offline)
$$\mathop {\lim }\limits_{x \to 0} {{\left( {1 - \cos 2x} \right)\left( {3 + \cos x} \right)} \over {x\tan 4x}}$$ is equal to
JEE Main 2015 (Offline)
If the function. $$g\left( x \right) = \left\{ {\matrix{ {k\sqrt {x + 1} ,} & {0 \le x \le 3} \cr {m\,x + 2,} & {3 < x \le 5} \cr ... JEE Main 2014 (Offline)$$\mathop {\lim }\limits_{x \to 0} {{\sin \left( {\pi {{\cos }^2}x} \right)} \over {{x^2}}}$$is equal to : JEE Main 2013 (Offline)$$\mathop {\lim }\limits_{x \to 0} {{\left( {1 - \cos 2x} \right)\left( {3 + \cos x} \right)} \over {x\tan 4x}}$$is equal to AIEEE 2012 If$$f:R \to R$$is a function defined by$$f\left( x \right) = \left[ x \right]\cos \left( {{{2x - 1} \over 2}} \right)\pi $$, where [x] denotes the ... AIEEE 2012 Consider the function,$$f\left( x \right) = \left| {x - 2} \right| + \left| {x - 5} \right|,x \in R$$Statement - 1 :$$f'\left( 4 \right) = 0$$Stat... AIEEE 2011$$\mathop {\lim }\limits_{x \to 2} \left( {{{\sqrt {1 - \cos \left\{ {2(x - 2)} \right\}} } \over {x - 2}}} \right)$$AIEEE 2011 The value of$$p$$and$$q$$for which the function$$f\left( x \right) = \left\{ {\matrix{ {{{\sin (p + 1)x + \sin x} \over x}} & {,x < 0} ...
AIEEE 2010
Let $$f:R \to R$$ be a positive increasing function with $$\mathop {\lim }\limits_{x \to \infty } {{f(3x)} \over {f(x)}} = 1$$. Then $$\mathop {\lim }... AIEEE 2009 Let$$f\left( x \right) = x\left| x \right|$$and$$g\left( x \right) = \sin x.$$Statement-1: gof is differentiable at$$x=0$$and its derivative is ... AIEEE 2008 Let$$f\left( x \right) = \left\{ {\matrix{ {\left( {x - 1} \right)\sin {1 \over {x - 1}}} & {if\,x \ne 1} \cr 0 & {if\,x = 1} \cr ...
AIEEE 2007
Let $$f:R \to R$$ be a function defined by $$f(x) = \min \left\{ {x + 1,\left| x \right| + 1} \right\}$$, then which of the following is true?
AIEEE 2007
The function $$f:R/\left\{ 0 \right\} \to R$$ given by $$f\left( x \right) = {1 \over x} - {2 \over {{e^{2x}} - 1}}$$ can be made continuous at $$x$$ ...
AIEEE 2006
The set of points where $$f\left( x \right) = {x \over {1 + \left| x \right|}}$$ is differentiable is
AIEEE 2005
Suppose $$f(x)$$ is differentiable at x = 1 and $$\mathop {\lim }\limits_{h \to 0} {1 \over h}f\left( {1 + h} \right) = 5$$, then $$f'\left( 1 \right)... AIEEE 2005 Let$$\alpha$$and$$\beta$$be the distinct roots of$$a{x^2} + bx + c = 0$$, then$$\mathop {\lim }\limits_{x \to \alpha } {{1 - \cos \left( {a{x^2}...
AIEEE 2005
If $$f$$ is a real valued differentiable function satisfying $$\left| {f\left( x \right) - f\left( y \right)} \right|$$ $$\le {\left( {x - y} \right)... AIEEE 2004 Let$$f(x) = {{1 - \tan x} \over {4x - \pi }}$$,$$x \ne {\pi \over 4}$$,$$x \in \left[ {0,{\pi \over 2}} \right]$$. If$$f(x)$$is continuous in ... AIEEE 2004 If$$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} + {b \over {{x^2}}}} \right)^{2x}} = {e^2}$$, then the value of$$a$$and$$b$$, ... AIEEE 2003 If$$\mathop {\lim }\limits_{x \to 0} {{\log \left( {3 + x} \right) - \log \left( {3 - x} \right)} \over x}$$= k, the value of k is AIEEE 2003 Let$$f(a) = g(a) = k$$and their nth derivatives$${f^n}(a)$$,$${g^n}(a)$$exist and are not equal for some n. Further if$$\mathop {\lim }\limits_...
AIEEE 2003
$$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{\left[ {1 - \tan \left( {{x \over 2}} \right)} \right]\left[ {1 - \sin x} \right]} \over {\left[ {1 ... AIEEE 2003 If$$f(x) = \left\{ {\matrix{ {x{e^{ - \left( {{1 \over {\left| x \right|}} + {1 \over x}} \right)}}} & {,x \ne 0} \cr 0 & {,x = 0} \...
AIEEE 2002
$$\mathop {\lim }\limits_{x \to 0} {{\sqrt {1 - \cos 2x} } \over {\sqrt 2 x}}$$ is
AIEEE 2002
$$\mathop {\lim }\limits_{x \to \infty } {\left( {{{{x^2} + 5x + 3} \over {{x^2} + x + 2}}} \right)^x}$$
AIEEE 2002
Let $$f(2) = 4$$ and $$f'(x) = 4.$$ Then $$\mathop {\lim }\limits_{x \to 2} {{xf\left( 2 \right) - 2f\left( x \right)} \over {x - 2}}$$ is given by...
AIEEE 2002
$$\mathop {\lim }\limits_{x \to 0} {{\log {x^n} - \left[ x \right]} \over {\left[ x \right]}}$$, $$n \in N$$, ( [x] denotes the greatest integer less ...
AIEEE 2002
If $$f\left( 1 \right) = 1,{f^1}\left( 1 \right) = 2,$$ then $$\mathop {\lim }\limits_{x \to 1} {{\sqrt {f\left( x \right)} - 1} \over {\sqrt x - 1}... AIEEE 2002$$f$$is defined in$$\left[ { - 5,5} \right]$$as$$f\left( x \right) = x$$if$$x$$is rational$$\,\,\,\,\,\,\,\,\,\,\,\,\, = - x$$if$$x$$... AIEEE 2002 f(x) and g(x) are two differentiable functions on [0, 2] such that f''(x) - g''(x) = 0, f'(1) = 2, g'(1) = 4, f(2) = 3, g(2) = 9 then f(x) - g(x) at x... AIEEE 2002 If f(x + y) = f(x).f(y)$$\forall  x, y and f(5) = 2, f'(0) = 3, then f'(5) is
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