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JEE Main

Limits, Continuity and Differentiability

Mathematics

Previous Years Questions

Numerical

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$$\lim\limits_{x \rightarrow 0}\left(\frac{(x+2 \cos x)^{3}+2(x+2 \cos x)^{2}+3 \sin (x+2 \cos x)}{(x+2)^{3}+2(x+2)^{2}+...
JEE Main 2022 (Online) 28th July Morning Shift
$$ \begin{aligned} &\text { If } \lim _{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(n k+1)+(n k+2)+\ldots+(n k+n)...
JEE Main 2022 (Online) 25th July Morning Shift
Suppose $$\mathop {\lim }\limits_{x \to 0} {{F(x)} \over {{x^3}}}$$ exists and is equal to L, where $$F(x) = \left| {\ma...
JEE Main 2022 (Online) 30th June Morning Shift
Let $$f(x) = \left[ {2{x^2} + 1} \right]$$ and $$g(x) = \left\{ {\matrix{ {2x - 3,} & {x ...
JEE Main 2022 (Online) 25th June Evening Shift
If $$\mathop {\lim }\limits_{x \to 1} {{\sin (3{x^2} - 4x + 1) - {x^2} + 1} \over {2{x^3} - 7{x^2} + ax + b}} = - 2$$, ...
JEE Main 2022 (Online) 28th June Evening Shift
Let [t] denote the greatest integer $$\le$$ t. The number of points where the function $$f(x) = [x]\left| {{x^2} - 1} \r...
JEE Main 2021 (Online) 1st September Evening Shift
Let $$f(x) = {x^6} + 2{x^4} + {x^3} + 2x + 3$$, x $$\in$$ R. Then the natural number n for which $$\mathop {\lim }\limit...
JEE Main 2021 (Online) 1st September Evening Shift
Let a, b $$\in$$ R, b $$\in$$ 0, Define a function $$f(x) = \left\{ {\matrix{ {a\sin {\pi \over 2}(x - 1),} & {f...
JEE Main 2021 (Online) 26th August Morning Shift
Let $$f:[0,3] \to R$$ be defined by $$f(x) = \min \{ x - [x],1 + [x] - x\} $$ where [x] is the greatest integer less tha...
JEE Main 2021 (Online) 27th July Morning Shift
Consider the function where P(x) is a polynomial such that P'' (x) is always a constant and P(3) = 9. If f(x) i...
JEE Main 2021 (Online) 25th July Evening Shift
Let f : R $$\to$$ R be a function defined as $$f(x) = \left\{ {\matrix{ {3\left( {1 - {{|x|} \over 2}} \right)} &...
JEE Main 2021 (Online) 22th July Evening Shift
If $$\mathop {\lim }\limits_{x \to 0} {{\alpha x{e^x} - \beta {{\log }_e}(1 + x) + \gamma {x^2}{e^{ - x}}} \over {x{{\si...
JEE Main 2021 (Online) 20th July Evening Shift
Let a function g : [ 0, 4 ] $$\to$$ R be defined as $$g(x) = \left\{ {\matrix{ {\mathop {\max }\limits_{0 \le t \le x...
JEE Main 2021 (Online) 20th July Evening Shift
If the value of $$\mathop {\lim }\limits_{x \to 0} {(2 - \cos x\sqrt {\cos 2x} )^{\left( {{{x + 2} \over {{x^2}}}} \righ...
JEE Main 2021 (Online) 20th July Morning Shift
Let f : R $$ \to $$ R satisfy the equation f(x + y) = f(x) . f(y) for all x, y $$\in$$R and f(x) $$\ne$$ 0 for any x$$\i...
JEE Main 2021 (Online) 18th March Evening Shift
If the function $$f(x) = {{\cos (\sin x) - \cos x} \over {{x^4}}}$$ is continuous at each point in its domain and $$f(0)...
JEE Main 2021 (Online) 17th March Morning Shift
Let f : R $$ \to $$ R and g : R $$ \to $$ R be defined as $$f(x) = \left\{ {\matrix{ {x + a,} & {x < 0} \cr ...
JEE Main 2021 (Online) 16th March Evening 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 ...
JEE Main 2021 (Online) 16th March Morning Shift
Let f : (0, 2) $$ \to $$ R be defined as f(x) = log2$$\left( {1 + \tan \left( {{{\pi x} \over 4}} \right)} \right)$$. Th...
JEE Main 2021 (Online) 16th March Morning Shift
If $$\mathop {\lim }\limits_{x \to 0} {{ax - ({e^{4x}} - 1)} \over {ax({e^{4x}} - 1)}}$$ exists and is equal to b, then ...
JEE Main 2021 (Online) 25th February Evening Shift
A function f is defined on [$$-$$3, 3] as$$f(x) = \left\{ {\matrix{ {\min \{ |x|,2 - {x^2}\} ,} & { - 2 \le x \le...
JEE Main 2021 (Online) 25th February Evening Shift
Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x6 is unity and it has extrema at x = $$-$$1 and ...
JEE Main 2021 (Online) 25th February Morning Slot
The number of points, at which the function f(x) = | 2x + 1 | $$-$$ 3| x + 2 | + | x2 + x $$-$$ 2 |, x$$\in$$R is not di...
JEE Main 2021 (Online) 25th February Morning Slot
$$\mathop {\lim }\limits_{n \to \infty } \tan \left\{ {\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {{1 \over {1 + r + ...
JEE Main 2021 (Online) 24th February Morning Slot
Let f : R $$ \to $$ R be defined as $$f\left( x \right) = \left\{ {\matrix{ {{x^5}\sin \left( {{1 \over x}} \right) +...
JEE Main 2020 (Online) 6th September Morning Slot
Let $$f(x) = x.\left[ {{x \over 2}} \right]$$, for -10< x < 10, where [t] denotes the greatest integer function. ...
JEE Main 2020 (Online) 5th September Morning Slot
Suppose a differentiable function f(x) satisfies the identity f(x+y) = f(x) + f(y) + xy2 + x2y, for all real x and y. $$...
JEE Main 2020 (Online) 4th September Morning Slot
If $$\mathop {\lim }\limits_{x \to 0} \left\{ {{1 \over {{x^8}}}\left( {1 - \cos {{{x^2}} \over 2} - \cos {{{x^2}} \over...
JEE Main 2020 (Online) 3rd September Morning Slot
If $$\mathop {\lim }\limits_{x \to 1} {{x + {x^2} + {x^3} + ... + {x^n} - n} \over {x - 1}}$$ = 820, (n $$ \in $$ N) the...
JEE Main 2020 (Online) 2nd September Morning Slot
If the function ƒ defined on $$\left( { - {1 \over 3},{1 \over 3}} \right)$$ by f(x) = $$\left\{ {\matrix{ {{1 \over ...
JEE Main 2020 (Online) 7th January Evening Slot
$$\mathop {\lim }\limits_{x \to 2} {{{3^x} + {3^{3 - x}} - 12} \over {{3^{ - x/2}} - {3^{1 - x}}}}$$ is equal to_______...
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...
JEE Main 2020 (Online) 7th January Morning Slot

MCQ (Single Correct Answer)

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If for $$\mathrm{p} \neq \mathrm{q} \neq 0$$, the function $$f(x)=\frac{\sqrt[7]{\mathrm{p}(729+x)}-3}{\sqrt[3]{729+\mat...
JEE Main 2022 (Online) 27th July Evening Shift
Let $$\beta=\mathop {\lim }\limits_{x \to 0} \frac{\alpha x-\left(e^{3 x}-1\right)}{\alpha x\left(e^{3 x}-1\right)}$$ fo...
JEE Main 2022 (Online) 26th July Evening Shift
If $$f(x) = \left\{ {\matrix{ {x + a} & , & {x \le 0} \cr {|x - 4|} & , & {x > 0} \cr } } \right.$$ and $$g(...
JEE Main 2022 (Online) 26th July Morning Shift
If the function $$f(x) = \left\{ {\matrix{ {{{{{\log }_e}(1 - x + {x^2}) + {{\log }_e}(1 + x + {x^2})} \over {\sec x ...
JEE Main 2022 (Online) 26th July Morning Shift
$$\mathop {\lim }\limits_{n \to \infty } {1 \over {{2^n}}}\left( {{1 \over {\sqrt {1 - {1 \over {{2^n}}}} }} + {1 \over ...
JEE Main 2022 (Online) 25th July Evening Shift
$$\lim _{x \rightarrow \frac{\pi}{4}} \frac{8 \sqrt{2}-(\cos x+\sin x)^{7}}{\sqrt{2}-\sqrt{2} \sin 2 x}$$ is equal to ...
JEE Main 2022 (Online) 25th July Evening Shift
If $$\mathop {\lim }\limits_{n \to \infty } \left( {\sqrt {{n^2} - n - 1} + n\alpha + \beta } \right) = 0$$, then $$8(...
JEE Main 2022 (Online) 25th July Morning Shift
$$\mathop {\lim }\limits_{n \to \infty } \left( {{{{n^2}} \over {({n^2} + 1)(n + 1)}} + {{{n^2}} \over {({n^2} + 4)(n + ...
JEE Main 2022 (Online) 24th June Evening Shift
Let $$f(x) = \left\{ {\matrix{ {{{\sin (x - [x])} \over {x - [x]}}} & {,\,x \in ( - 2, - 1)} \cr {\max \{ 2x,3[|...
JEE Main 2022 (Online) 24th June Evening Shift
$$\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{{\tan }^2}x\left( {{{(2{{\sin }^2}x + 3\sin x + 4)}^{{1 \over 2...
JEE Main 2022 (Online) 25th 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 differentia...
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, g : R $$\to$$ R be two real valued functions defined as $$f(x) = \left\{ {\matrix{ { - |x + 3|} & , & {x 1 and...
JEE Main 2022 (Online) 26th June Morning Shift
$$\mathop {\lim }\limits_{x \to {1 \over {\sqrt 2 }}} {{\sin ({{\cos }^{ - 1}}x) - x} \over {1 - \tan ({{\cos }^{ - 1}}x...
JEE Main 2022 (Online) 26th 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] i...
JEE Main 2022 (Online) 27th June Morning Shift
Let f : R $$\to$$ R be defined as $$f(x) = \left[ {\matrix{ {[{e^x}],} & {x where a, b, c $$\in$$ R and [t] denotes ...
JEE Main 2022 (Online) 28th June Morning Shift
The value of $$\mathop {\lim }\limits_{n \to \infty } 6\tan \left\{ {\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {{1 \...
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_{x \to 1} {{({x^2} - 1){{\sin }^2}(\pi x)} \over {{x^4} - 2{x^3} + 2x - 1}}$$ is e...
JEE Main 2022 (Online) 29th June Evening Shift
Let f be any continuous function on [0, 2] and twice differentiable on (0, 2). If f(0) = 0, f(1) = 1 and f(2) = 2, then
JEE Main 2021 (Online) 31st August Evening Shift
If $$\alpha = \mathop {\lim }\limits_{x \to {\pi \over 4}} {{{{\tan }^3}x - \tan x} \over {\cos \left( {x + {\pi \ove...
JEE Main 2021 (Online) 31st August Evening 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) 31st August Morning Shift
If the function $$f(x) = \left\{ {\matrix{ {{1 \over x}{{\log }_e}\left( {{{1 + {x \over a}} \over {1 - {x \over b}}}...
JEE Main 2021 (Online) 31st August Morning Shift
The function $$f(x) = \left| {{x^2} - 2x - 3} \right|\,.\,{e^{\left| {9{x^2} - 12x + 4} \right|}}$$ is not differentiabl...
JEE Main 2021 (Online) 31st August Morning Shift
If $$\mathop {\lim }\limits_{x \to \infty } \left( {\sqrt {{x^2} - x + 1} - ax} \right) = b$$, then the ordered pair (a...
JEE Main 2021 (Online) 27th August Evening Shift
If $$\alpha$$, $$\beta$$ are the distinct roots of x2 + bx + c = 0, then $$\mathop {\lim }\limits_{x \to \beta } {{{e^{2...
JEE Main 2021 (Online) 27th August Morning Shift
$$\mathop {\lim }\limits_{x \to 2} \left( {\sum\limits_{n = 1}^9 {{x \over {n(n + 1){x^2} + 2(2n + 1)x + 4}}} } \right)$...
JEE Main 2021 (Online) 26th August Evening Shift
The value of $$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\sum\limits_{r = 0}^{2n - 1} {{{{n^2}} \over {{n^2} + 4...
JEE Main 2021 (Online) 26th August Morning Shift
Let f : R $$\to$$ R be a function such that f(2) = 4 and f'(2) = 1. Then, the value of $$\mathop {\lim }\limits_{x \to 2...
JEE Main 2021 (Online) 27th July Morning Shift
Let $$f:\left( { - {\pi \over 4},{\pi \over 4}} \right) \to R$$ be defined as $$f(x) = \left\{ {\matrix{ {{{(1 + |\...
JEE Main 2021 (Online) 27th July Morning Shift
Let $$f:[0,\infty ) \to [0,3]$$ be a function defined by $$f(x) = \left\{ {\matrix{ {\max \{ \sin t:0 \le t \le x\} ,...
JEE Main 2021 (Online) 27th July Evening Shift
The value of $$\mathop {\lim }\limits_{x \to 0} \left( {{x \over {\root 8 \of {1 - \sin x} - \root 8 \of {1 + \sin x} }...
JEE Main 2021 (Online) 27th July Evening Shift
Let f : R $$\to$$ R be defined as$$f(x) = \left\{ {\matrix{ {{{\lambda \left| {{x^2} - 5x + 6} \right|} \over {\mu (5...
JEE Main 2021 (Online) 25th July Morning Shift
Let $$f(x) = 3{\sin ^4}x + 10{\sin ^3}x + 6{\sin ^2}x - 3$$, $$x \in \left[ { - {\pi \over 6},{\pi \over 2}} \right]$$...
JEE Main 2021 (Online) 25th July Morning Shift
Let f : R $$\to$$ R be defined as $$f(x) = \left\{ {\matrix{ {{{{x^3}} \over {{{(1 - \cos 2x)}^2}}}{{\log }_e}\left( ...
JEE Main 2021 (Online) 22th July Evening Shift
Let f : R $$\to$$ R be defined as$$f(x) = \left\{ {\matrix{ { - {4 \over 3}{x^3} + 2{x^2} + 3x,} & {x > 0} \c...
JEE Main 2021 (Online) 22th July Evening Shift
The sum of all the local minimum values of the twice differentiable function f : R $$\to$$ R defined by $$f(x) = {x^3} -...
JEE Main 2021 (Online) 20th July Evening Shift
If $$f:R \to R$$ is given by $$f(x) = x + 1$$, then the value of $$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\le...
JEE Main 2021 (Online) 20th July Evening Shift
Let a function f : R $$\to$$ R be defined as $$f(x) = \left\{ {\matrix{ {\sin x - {e^x}} & {if} & {x \le 0} ...
JEE Main 2021 (Online) 20th July Morning Shift
Let f : R $$ \to $$ R be a function defined as$$f(x) = \left\{ \matrix{ {{\sin (a + 1)x + \sin 2x} \over {2x}},if\,x &...
JEE Main 2021 (Online) 18th March Evening Shift
If $$f(x) = \left\{ {\matrix{ {{1 \over {|x|}}} & {;\,|x|\, \ge 1} \cr {a{x^2} + b} & {;\,|x|\, < 1} ...
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...
JEE Main 2021 (Online) 18th March Morning Shift
The value of $$\mathop {\lim }\limits_{n \to \infty } {{[r] + [2r] + ... + [nr]} \over {{n^2}}}$$, where r is a non-zero...
JEE Main 2021 (Online) 17th March Evening Shift
The value of the limit $$\mathop {\lim }\limits_{\theta \to 0} {{\tan (\pi {{\cos }^2}\theta )} \over {\sin (2\pi {{\si...
JEE Main 2021 (Online) 17th March Evening Shift
The value of $$\mathop {\lim }\limits_{x \to {0^ + }} {{{{\cos }^{ - 1}}(x - {{[x]}^2}).{{\sin }^{ - 1}}(x - {{[x]}^2})}...
JEE Main 2021 (Online) 17th March Morning Shift
Let $$\alpha$$ $$\in$$ R be such that the function $$f(x) = \left\{ {\matrix{ {{{{{\cos }^{ - 1}}(1 - {{\{ x\} }^2}){...
JEE Main 2021 (Online) 16th March Evening Shift
Let f : S $$ \to $$ S where S = (0, $$\infty $$) be a twice differentiable function such that f(x + 1) = xf(x). If g : S...
JEE Main 2021 (Online) 16th March Evening Shift
Let the functions f : R $$ \to $$ R and g : R $$ \to $$ R be defined as :$$f(x) = \left\{ {\matrix{ {x + 2,} & {x...
JEE Main 2021 (Online) 16th March Morning Shift
Let $${S_k} = \sum\limits_{r = 1}^k {{{\tan }^{ - 1}}\left( {{{{6^r}} \over {{2^{2r + 1}} + {3^{2r + 1}}}}} \right)} $$....
JEE Main 2021 (Online) 16th March Morning Shift
Let f : R $$ \to $$ R be defined as $$f(x) = \left\{ \matrix{ 2\sin \left( { - {{\pi x} \over 2}} \right),if\,x < ...
JEE Main 2021 (Online) 26th February Evening Shift
Let $$f(x) = {\sin ^{ - 1}}x$$ and $$g(x) = {{{x^2} - x - 2} \over {2{x^2} - x - 6}}$$. If $$g(2) = \mathop {\lim }\limi...
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 $$\mathop {\lim }\limits_{x \to a} {{xf...
JEE Main 2021 (Online) 26th February Evening Shift
Let f be any function defined on R and let it satisfy the condition : $$|f(x) - f(y)|\, \le \,|{(x - y)^2}|,\forall (x,y...
JEE Main 2021 (Online) 26th February Morning Shift
The value of $$\mathop {\lim }\limits_{h \to 0} 2\left\{ {{{\sqrt 3 \sin \left( {{\pi \over 6} + h} \right) - \cos \lef...
JEE Main 2021 (Online) 26th February Morning Shift
$$\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over n} + {n \over {{{(n + 1)}^2}}} + {n \over {{{(n + 2)}^2}}} + ....
JEE Main 2021 (Online) 25th February Evening Shift
$$\mathop {\lim }\limits_{n \to \infty } {\left( {1 + {{1 + {1 \over 2} + ........ + {1 \over n}} \over {{n^2}}}} \right...
JEE Main 2021 (Online) 25th February Morning Slot
If f : R $$ \to $$ R is a function defined by f(x)= [x - 1] $$\cos \left( {{{2x - 1} \over 2}} \right)\pi $$, where [.] ...
JEE Main 2021 (Online) 24th February Morning Slot
For all twice differentiable functions f : R $$ \to $$ R, with f(0) = f(1) = f'(0) = 0
JEE Main 2020 (Online) 6th September Evening Slot
Let f : R $$ \to $$ R be a function defined by f(x) = max {x, x2}. Let S denote the set of all points in R, where f is n...
JEE Main 2020 (Online) 6th September Evening Slot
$$\mathop {\lim }\limits_{x \to 1} \left( {{{\int\limits_0^{{{\left( {x - 1} \right)}^2}} {t\cos \left( {{t^2}} \right)d...
JEE Main 2020 (Online) 6th September Morning Slot
$$\mathop {\lim }\limits_{x \to 0} {{x\left( {{e^{\left( {\sqrt {1 + {x^2} + {x^4}} - 1} \right)/x}} - 1} \right)} \ove...
JEE Main 2020 (Online) 5th September Evening Slot
If $$\alpha $$ is positive root of the equation, p(x) = x2 - x - 2 = 0, then $$\mathop {\lim }\limits_{x \to {\alpha ^ +...
JEE Main 2020 (Online) 5th September Morning Slot
The function $$f(x) = \left\{ {\matrix{ {{\pi \over 4} + {{\tan }^{ - 1}}x,} & {\left| x \right| \le 1} \cr ...
JEE Main 2020 (Online) 4th September Evening Slot
Let $$f:\left( {0,\infty } \right) \to \left( {0,\infty } \right)$$ be a differentiable function such that f(1) = e and ...
JEE Main 2020 (Online) 4th September Evening Slot
$$\mathop {\lim }\limits_{x \to a} {{{{\left( {a + 2x} \right)}^{{1 \over 3}}} - {{\left( {3x} \right)}^{{1 \over 3}}}} ...
JEE Main 2020 (Online) 3rd September Evening Slot
Let [t] denote the greatest integer $$ \le $$ t. If for some $$\lambda $$ $$ \in $$ R - {1, 0}, $$\mathop {\lim }\limits...
JEE Main 2020 (Online) 3rd September Morning Slot
$$\mathop {\lim }\limits_{x \to 0} {\left( {\tan \left( {{\pi \over 4} + x} \right)} \right)^{{1 \over x}}}$$ is equal ...
JEE Main 2020 (Online) 2nd September Evening Slot
If a function f(x) defined by $$f\left( x \right) = \left\{ {\matrix{ {a{e^x} + b{e^{ - x}},} & { - 1 \le x < ...
JEE Main 2020 (Online) 2nd September Morning Slot
Let a function ƒ : [0, 5] $$ \to $$ R be continuous, ƒ(1) = 3 and F be defined as : $$F(x) = \int\limits_1^x {{t^2}g(t)d...
JEE Main 2020 (Online) 9th January Evening Slot
Let [t] denote the greatest integer $$ \le $$ t and $$\mathop {\lim }\limits_{x \to 0} x\left[ {{4 \over x}} \right] = A...
JEE Main 2020 (Online) 9th January Evening Slot
If $$f(x) = \left\{ {\matrix{ {{{\sin (a + 2)x + \sin x} \over x};} & {x < 0} \cr {b\,\,\,\,\,\,\,\,\,\,\...
JEE Main 2020 (Online) 9th January Morning Slot
Let ƒ be any function continuous on [a, b] and twice differentiable on (a, b). If for all x $$ \in $$ (a, b), ƒ'(x) >...
JEE Main 2020 (Online) 9th January Morning Slot
$$\mathop {\lim }\limits_{x \to 0} {{\int_0^x {t\sin \left( {10t} \right)dt} } \over x}$$ is equal to
JEE Main 2020 (Online) 8th January Evening Slot
Let S be the set of all functions ƒ : [0,1] $$ \to $$ R, which are continuous on [0,1] and differentiable on (0,1). Then...
JEE Main 2020 (Online) 8th January Evening Slot
$$\mathop {\lim }\limits_{x \to 0} {\left( {{{3{x^2} + 2} \over {7{x^2} + 2}}} \right)^{{1 \over {{x^2}}}}}$$ is equal t...
JEE Main 2020 (Online) 8th January Morning Slot
Let ƒ(x) be a polynomial of degree 5 such that x = ±1 are its critical points. If $$\mathop {\lim }\limits_{x \to 0} \le...
JEE Main 2020 (Online) 7th January Evening Slot
The value of c in the Lagrange's mean value theorem for the function ƒ(x) = x3 - 4x2 + 8x + 11, when x $$ \in $$ [0, 1...
JEE Main 2020 (Online) 7th January Evening Slot
Let the function, ƒ:[-7, 0]$$ \to $$R be continuous on [-7,0] and differentiable on (-7, 0). If ƒ(-7) = - 3 and ƒ'(x) $$...
JEE Main 2020 (Online) 7th January Morning Slot
Let f(x) = 5 – |x – 2| and g(x) = |x + 1|, x $$ \in $$ R. If f(x) attains maximum value at $$\alpha $$ and g(x) attains ...
JEE Main 2019 (Online) 12th April Evening Slot
$$\mathop {\lim }\limits_{x \to 0} {{x + 2\sin x} \over {\sqrt {{x^2} + 2\sin x + 1} - \sqrt {{{\sin }^2}x - x + 1} }}$...
JEE Main 2019 (Online) 12th April Evening Slot
If $$\alpha $$ and $$\beta $$ are the roots of the equation 375x2 – 25x – 2 = 0, then $$\mathop {\lim }\limits_{n \to \i...
JEE Main 2019 (Online) 12th April Morning Slot
If $$\mathop {\lim }\limits_{x \to 1} {{{x^2} - ax + b} \over {x - 1}} = 5$$, then a + b is equal to :
JEE Main 2019 (Online) 10th April Evening Slot
$$\mathop {\lim }\limits_{n \to \infty } \left( {{{{{(n + 1)}^{1/3}}} \over {{n^{4/3}}}} + {{{{(n + 2)}^{1/3}}} \over {{...
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$$f(x) = \left\{ {\matrix{ {{{\sin (p + 1)x + \sin x} \over x}} & {,x < 0} \cr q & {,x = 0} \cr ...
JEE Main 2019 (Online) 10th April Morning Slot
If $$\mathop {\lim }\limits_{x \to 1} {{{x^4} - 1} \over {x - 1}} = \mathop {\lim }\limits_{x \to k} {{{x^3} - {k^3}} \o...
JEE Main 2019 (Online) 10th April Morning Slot
If the function $$f(x) = \left\{ {\matrix{ {a|\pi - x| + 1,x \le 5} \cr {b|x - \pi | + 3,x > 5} \cr } } ...
JEE Main 2019 (Online) 9th April Evening Slot
If $$f(x) = [x] - \left[ {{x \over 4}} \right]$$ ,x $$ \in $$ 4 , where [x] denotes the greatest integer function, then
JEE Main 2019 (Online) 9th April Evening Slot
If the function ƒ defined on , $$\left( {{\pi \over 6},{\pi \over 3}} \right)$$ by $$$f(x) = \left\{ {\matrix{ {{{\...
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 d...
JEE Main 2019 (Online) 9th April Morning Slot
Let ƒ : [–1,3] $$ \to $$ R be defined as $$f(x) = \left\{ {\matrix{ {\left| x \right| + \left[ x \right]} & , &am...
JEE Main 2019 (Online) 8th April Evening Slot
Let ƒ : R $$ \to $$ R be a differentiable function satisfying ƒ'(3) + ƒ'(2) = 0. Then $$\mathop {\lim }\limits_{x \to 0}...
JEE Main 2019 (Online) 8th April Evening Slot
$$\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}x} \over {\sqrt 2 - \sqrt {1 + \cos x} }}$$ equals:
JEE Main 2019 (Online) 8th April Morning 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...
JEE Main 2019 (Online) 12th January Evening Slot
$$\mathop {\lim }\limits_{x \to \infty } \left( {{n \over {{n^2} + {1^2}}} + {n \over {{n^2} + {2^2}}} + {n \over {{n^2}...
JEE Main 2019 (Online) 12th January Evening Slot
$$\mathop {\lim }\limits_{x \to {1^ - }} {{\sqrt \pi - \sqrt {2{{\sin }^{ - 1}}x} } \over {\sqrt {1 - x} }}$$ is equal...
JEE Main 2019 (Online) 12th January Evening Slot
Let S be the set of all points in (–$$\pi $$, $$\pi $$) at which the function, f(x) = min{sin x, cos x} is not different...
JEE Main 2019 (Online) 12th January Morning Slot
$$\mathop {\lim }\limits_{x \to \pi /4} {{{{\cot }^3}x - \tan x} \over {\cos \left( {x + {\pi \over 4}} \right)}}$$ is ...
JEE Main 2019 (Online) 12th January Morning Slot
Let f and g be continuous functions on [0, a] such that f(x) = f(a – x) and g(x) + g(a – x) = 4, then $$\int\limits_0^a ...
JEE Main 2019 (Online) 12th January Morning Slot
$$\mathop {\lim }\limits_{x \to 0} {{x\cot \left( {4x} \right)} \over {{{\sin }^2}x{{\cot }^2}\left( {2x} \right)}}$$ is...
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 differ...
JEE Main 2019 (Online) 11th January Evening Slot
Let [x] denote the greatest integer less than or equal to x. Then $$\mathop {\lim }\limits_{x \to 0} {{\tan \left( {\pi ...
JEE Main 2019 (Online) 11th January Morning Slot
Let $$f\left( x \right) = \left\{ {\matrix{ { - 1} & { - 2 \le x < 0} \cr {{x^2} - 1,} & {0 \le x \le...
JEE Main 2019 (Online) 11th January Morning Slot
Let f : ($$-$$1, 1) $$ \to $$ R be a function defined by f(x) = max $$\left\{ { - \left| x \right|, - \sqrt {1 - {x^2}} ...
JEE Main 2019 (Online) 10th January Evening Slot
Let  $$f\left( x \right) = \left\{ {\matrix{ {\max \left\{ {\left| x \right|,{x^2}} \right\}} & {\left|...
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  $$\mathop {\lim }\limit...
JEE Main 2019 (Online) 10th January Morning Slot
Let f : R $$ \to $$ R be a function such that f(x) = x3 + x2f'(1) + xf''(2) + f'''(3), x $$ \in $$ R. Then f(2) equals...
JEE Main 2019 (Online) 10th January Morning Slot
Let f be a differentiable function from R to R such that $$\left| {f\left( x \right) - f\left( y \right)} \right| \le 2...
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 $$\mathop {\lim }\limits_{x \to {0^...
JEE Main 2019 (Online) 9th January Evening Slot
$$\mathop {\lim }\limits_{y \to 0} {{\sqrt {1 + \sqrt {1 + {y^4}} } - \sqrt 2 } \over {{y^4}}}$$
JEE Main 2019 (Online) 9th January Morning Slot
Let f : R $$ \to $$ R be a function defined as $$f(x) = \left\{ {\matrix{ 5 & ; & {x \le 1} \cr {a + bx...
JEE Main 2019 (Online) 9th January 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 continuou...
JEE Main 2018 (Online) 16th April Morning Slot
If $$x = \sqrt {{2^{\cos e{c^{ - 1}}}}} $$ and $$y = \sqrt {{2^{se{c^{ - 1}}t}}} \,\,\left( {\left| t \right| \ge 1} \ri...
JEE Main 2018 (Online) 16th April Morning Slot
$$\mathop {\lim }\limits_{x \to 0} \,\,{{{{\left( {27 + x} \right)}^{{1 \over 3}}} - 3} \over {9 - {{\left( {27 + x} \ri...
JEE Main 2018 (Online) 16th April Morning Slot
Let f(x) be a polynomial of degree $$4$$ having extreme values at $$x = 1$$ and $$x = 2.$$ If   $$\mathop {lim}\lim...
JEE Main 2018 (Online) 15th April Evening Slot
If    f(x) = sin-1 $$\left( {{{2 \times {3^x}} \over {1 + {9^x}}}} \right),$$ then f'$$\left( { - {1 \over 2}}...
JEE Main 2018 (Online) 15th April Evening Slot
Let f(x) = $$\left\{ {\matrix{ {{{\left( {x - 1} \right)}^{{1 \over {2 - x}}}},} & {x > 1,x \ne 2} \cr {k...
JEE Main 2018 (Online) 15th April Evening Slot
$$\mathop {\lim }\limits_{x \to 0} {{x\tan 2x - 2x\tan x} \over {{{\left( {1 - \cos 2x} \right)}^2}}}$$ equals :
JEE Main 2018 (Online) 15th April Evening Slot
Let S = {($$\lambda $$, $$\mu $$) $$ \in $$ R $$ \times $$ R : f(t) = (|$$\lambda $$| e|t| $$-$$ $$\mu $$). sin (2|t|),...
JEE Main 2018 (Online) 15th April Morning Slot
If   x2 + y2 + sin y = 4, then the value of $${{{d^2}y} \over {d{x^2}}}$$ at the point ($$-$$2,0) is :...
JEE Main 2018 (Online) 15th April Morning Slot
If $$f\left( x \right) = \left| {\matrix{ {\cos x} & x & 1 \cr {2\sin x} & {{x^2}} & {2x} \cr ...
JEE Main 2018 (Online) 15th April Morning Slot
For each t $$ \in R$$, let [t] be the greatest integer less than or equal to t. Then $$\mathop {\lim }\limits_{x \to {0^...
JEE Main 2018 (Offline)
If 2x = y$${^{{1 \over 5}}}$$ + y$${^{ - {1 \over 5}}}$$ and (x2 $$-$$ 1) $${{{d^2}y} \over {d{x^2}}}$$ + $$\lambda $$x...
JEE Main 2017 (Online) 9th April Morning Slot
The value of k for which the function $$f\left( x \right) = \left\{ {\matrix{ {{{\left( {{4 \over 5}} \right)}^{{{\t...
JEE Main 2017 (Online) 9th April Morning Slot
Let f be a polynomial function such that f (3x) = f ' (x) . f '' (x), for all x $$ \in $$ R. Then :
JEE Main 2017 (Online) 9th April Morning Slot
If y = $${\left[ {x + \sqrt {{x^2} - 1} } \right]^{15}} + {\left[ {x - \sqrt {{x^2} - 1} } \right]^{15}},$$ then (x2 $$...
JEE Main 2017 (Online) 8th April Morning Slot
$$\mathop {\lim }\limits_{x \to 3} $$ $${{\sqrt {3x} - 3} \over {\sqrt {2x - 4} - \sqrt 2 }}$$ is equal to :
JEE Main 2017 (Online) 8th April Morning Slot
If for $$x \in \left( {0,{1 \over 4}} \right)$$, the derivatives of $${\tan ^{ - 1}}\left( {{{6x\sqrt x } \over {1 - 9{x...
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 2017 (Offline)
Let a, b $$ \in $$ R, (a $$ \ne $$ 0). If the function f defined as $$f\left( x \right) = \left\{ {\matrix{ {{{2{x^2}...
JEE Main 2016 (Online) 10th April Morning Slot
$$\mathop {\lim }\limits_{x \to 0} \,{{{{\left( {1 - \cos 2x} \right)}^2}} \over {2x\,\tan x\, - x\tan 2x}}$$ is :
JEE Main 2016 (Online) 10th April Morning Slot
If the function f(x) = $$\left\{ {\matrix{ { - x} & {x < 1} \cr {a + {{\cos }^{ - 1}}\left( {x + b} \rig...
JEE Main 2016 (Online) 9th April Morning Slot
If    $$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} - {4 \over {{x^2}}}} \right)^{2x}} = {...
JEE Main 2016 (Online) 9th April Morning Slot
Let $$p = \mathop {\lim }\limits_{x \to {0^ + }} {\left( {1 + {{\tan }^2}\sqrt x } \right)^{{1 \over {2x}}}}$$ then $$lo...
JEE Main 2016 (Offline)
For $$x \in \,R,\,\,f\left( x \right) = \left| {\log 2 - \sin x} \right|\,\,$$ and $$\,\,g\left( x \right) = f\left( {f...
JEE Main 2016 (Offline)
$$\mathop {\lim }\limits_{n \to \infty } {\left( {{{\left( {n + 1} \right)\left( {n + 2} \right)...3n} \over {{n^{2n}}}}...
JEE Main 2016 (Offline)
If the function. $$g\left( x \right) = \left\{ {\matrix{ {k\sqrt {x + 1} ,} & {0 \le x \le 3} \cr {m\,x + 2,...
JEE Main 2015 (Offline)
$$\mathop {\lim }\limits_{x \to 0} {{\left( {1 - \cos 2x} \right)\left( {3 + \cos x} \right)} \over {x\tan 4x}}$$ is equ...
JEE Main 2015 (Offline)
$$\mathop {\lim }\limits_{x \to 0} {{\sin \left( {\pi {{\cos }^2}x} \right)} \over {{x^2}}}$$ is equal to :
JEE Main 2014 (Offline)
$$\mathop {\lim }\limits_{x \to 0} {{\left( {1 - \cos 2x} \right)\left( {3 + \cos x} \right)} \over {x\tan 4x}}$$ is equ...
JEE Main 2013 (Offline)
If $$f:R \to R$$ is a function defined by $$f\left( x \right) = \left[ x \right]\cos \left( {{{2x - 1} \over 2}} \right)...
AIEEE 2012
Consider the function, $$f\left( x \right) = \left| {x - 2} \right| + \left| {x - 5} \right|,x \in R$$ Statement - 1 : $...
AIEEE 2012
The value of $$p$$ and $$q$$ for which the function $$f\left( x \right) = \left\{ {\matrix{ {{{\sin (p + 1)x + \sin x...
AIEEE 2011
$$\mathop {\lim }\limits_{x \to 2} \left( {{{\sqrt {1 - \cos \left\{ {2(x - 2)} \right\}} } \over {x - 2}}} \right)$$
AIEEE 2011
Let $$f:R \to R$$ be a positive increasing function with $$\mathop {\lim }\limits_{x \to \infty } {{f(3x)} \over {f(x)}}...
AIEEE 2010
Let $$f\left( x \right) = \left\{ {\matrix{ {\left( {x - 1} \right)\sin {1 \over {x - 1}}} & {if\,x \ne 1} \cr ...
AIEEE 2008
The function $$f:R/\left\{ 0 \right\} \to R$$ given by $$f\left( x \right) = {1 \over x} - {2 \over {{e^{2x}} - 1}}$$ ca...
AIEEE 2007
Let $$f:R \to R$$ be a function defined by $$f(x) = \min \left\{ {x + 1,\left| x \right| + 1} \right\}$$, then which of ...
AIEEE 2007
If $$f$$ is a real valued differentiable function satisfying $$\left| {f\left( x \right) - f\left( y \right)} \right|$$ ...
AIEEE 2005
Let f be differentiable for all x. If f(1) = -2 and f'(x) $$ \ge $$ 2 for x $$ \in \left[ {1,6} \right]$$, then
AIEEE 2005
Let $$\alpha$$ and $$\beta$$ be the distinct roots of $$a{x^2} + bx + c = 0$$, then $$\mathop {\lim }\limits_{x \to \alp...
AIEEE 2005
Suppose $$f(x)$$ is differentiable at x = 1 and $$\mathop {\lim }\limits_{h \to 0} {1 \over h}f\left( {1 + h} \right) = ...
AIEEE 2005
$$\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over {{n^2}}}{{\sec }^2}{1 \over {{n^2}}} + {2 \over {{n^2}}}{{\sec...
AIEEE 2005
If $$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} + {b \over {{x^2}}}} \right)^{2x}} = {e^2}$$, then ...
AIEEE 2004
Let $$f(x) = {{1 - \tan x} \over {4x - \pi }}$$, $$x \ne {\pi \over 4}$$, $$x \in \left[ {0,{\pi \over 2}} \right]$$. ...
AIEEE 2004
If $$f(x) = \left\{ {\matrix{ {x{e^{ - \left( {{1 \over {\left| x \right|}} + {1 \over x}} \right)}}} & {,x \ne 0...
AIEEE 2003
$$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{\left[ {1 - \tan \left( {{x \over 2}} \right)} \right]\left[ {1 - \sin...
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...
AIEEE 2003
If $$\mathop {\lim }\limits_{x \to 0} {{\log \left( {3 + x} \right) - \log \left( {3 - x} \right)} \over x}$$ = k, the v...
AIEEE 2003
The value of $$\mathop {\lim }\limits_{x \to 0} {{\int\limits_0^{{x^2}} {{{\sec }^2}tdt} } \over xsinx}$$ is
AIEEE 2003
$$\mathop {\lim }\limits_{n \to \infty } {{1 + {2^4} + {3^4} + .... + {n^4}} \over {{n^5}}}$$ - $$\mathop {\lim }\limits...
AIEEE 2003
If f(x + y) = f(x).f(y) $$\forall $$ x, y and f(5) = 2, f'(0) = 3, then f'(5) is
AIEEE 2002
f(x) and g(x) are two differentiable functions on [0, 2] such that f''(x) - g''(x) = 0, f'(1) = 2, g'(1) = 4, f(2) = 3, ...
AIEEE 2002
$$f$$ is defined in $$\left[ { - 5,5} \right]$$ as $$f\left( x \right) = x$$ if $$x$$ is rational $$\,\,\,\,\,\,\,\,\,\,...
AIEEE 2002
If $$f\left( 1 \right) = 1,{f^1}\left( 1 \right) = 2,$$ then $$\mathop {\lim }\limits_{x \to 1} {{\sqrt {f\left( x \righ...
AIEEE 2002
$$\mathop {\lim }\limits_{x \to 0} {{\log {x^n} - \left[ x \right]} \over {\left[ x \right]}}$$, $$n \in N$$, ( [x] deno...
AIEEE 2002
$$\mathop {\lim }\limits_{x \to \infty } {\left( {{{{x^2} + 5x + 3} \over {{x^2} + x + 2}}} \right)^x}$$
AIEEE 2002
$$\mathop {\lim }\limits_{n \to \infty } {{{1^p} + {2^p} + {3^p} + ..... + {n^p}} \over {{n^{p + 1}}}}$$ is
AIEEE 2002
Let $$f(2) = 4$$ and $$f'(x) = 4.$$ Then $$\mathop {\lim }\limits_{x \to 2} {{xf\left( 2 \right) - 2f\left( x \right)} \...
AIEEE 2002
$$\mathop {\lim }\limits_{x \to 0} {{\sqrt {1 - \cos 2x} } \over {\sqrt 2 x}}$$ is
AIEEE 2002

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