JEE Main
Mathematics
Limits, Continuity and Differentiability
Previous Years Questions

$$\lim\limits_{x \rightarrow \infty} \frac{(\sqrt{3 x+1}+\sqrt{3 x-1})^6+(\sqrt{3 x+1}-\sqrt{3 x-1})^6}{\left(x+\sqrt{x^2-1}\right)^6+\left(x-\sqrt{x... \lim\limits_{n \rightarrow \infty} \frac{3}{n}\left\{4+\left(2+\frac{1}{n}\right)^2+\left(2+\frac{2}{n}\right)^2+\ldots+\left(3-\frac{1}{n}\right)^2\... Let f, g and h be the real valued functions defined on \mathbb{R} as f(x)=\left\{\begin{array}{cc}\frac{x}{|x|}, & x \neq 0 \\ 1, & x=0\end{arr... Let$$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)}...
If the function $$f(x) = \left\{ {\matrix{ {(1 + |\cos x|)^{\lambda \over {|\cos x|}}} & , & {0 is continuous at$$x = {\pi \over 2}$$, then$$9...
The value of $$\mathop {\lim }\limits_{n \to \infty } {{1 + 2 - 3 + 4 + 5 - 6\, + \,.....\, + \,(3n - 2) + (3n - 1) - 3n} \over {\sqrt {2{n^4} + 4n + ... The set of all values of$$a$$for which$$\mathop {\lim }\limits_{x \to a} ([x - 5] - [2x + 2]) = 0$$, where [$$\alpha$$] denotes the greatest intege...$$\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...
Let $$f(x) = \left\{ {\matrix{ {{x^2}\sin \left( {{1 \over x}} \right)} & {,\,x \ne 0} \cr 0 & {,\,x = 0} \cr } } \right.$$ Then at $$x=0...$$ \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 ...
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 $$\alpha, \bet... The number of points, where the function$$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|$$,... The function$$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+...
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...
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 $$\alpha \in \mathbb{R}... If the function$$f(x) = \left\{ {\matrix{ {{{{{\log }_e}(1 - x + {x^2}) + {{\log }_e}(1 + x + {x^2})} \over {\sec x - \cos x}}} & , & {x \in \left...
If $$f(x) = \left\{ {\matrix{ {x + a} & , & {x \le 0} \cr {|x - 4|} & , & {x > 0} \cr } } \right.$$ and $$g(x) = \left\{ {\matrix{ {x +...$$\lim\limits_{x \rightarrow \frac{\pi}{4}} \frac{8 \sqrt{2}-(\cos x+\sin x)^{7}}{\sqrt{2}-\sqrt{2} \sin 2 x}$$is equal to If$$\mathop {\lim }\limits_{n \to \infty } \left( {\sqrt {{n^2} - n - 1} + n\alpha + \beta } \right) = 0$$, then$$8(\alpha+\beta)$$is equal to : ... The value of$$\mathop {\lim }\limits_{x \to 1} {{({x^2} - 1){{\sin }^2}(\pi x)} \over {{x^4} - 2{x^3} + 2x - 1}}$$is equal to: Let f, g : R$$\to$$R be functions defined by$$f(x) = \left\{ {\matrix{ {[x]} & , & {x ...
The value of $$\mathop {\lim }\limits_{n \to \infty } 6\tan \left\{ {\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {{1 \over {{r^2} + 3r + 3}}} \right... Let f : R$$\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 ...
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. ...
$$\mathop {\lim }\limits_{x \to 0} {{\cos (\sin x) - \cos x} \over {{x^4}}}$$ is equal to :
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...
$$\mathop {\lim }\limits_{x \to {1 \over {\sqrt 2 }}} {{\sin ({{\cos }^{ - 1}}x) - x} \over {1 - \tan ({{\cos }^{ - 1}}x)}}$$ is equal to :
Let f, g : R $$\to$$ R be two real valued functions defined as $$f(x) = \left\{ {\matrix{ { - |x + 3|} & , & {x 1 and k2 are real constants. If (go...$$\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 ...
Let $$f(x) = \left\{ {\matrix{ {{{\sin (x - [x])} \over {x - [x]}}} & {,\,x \in ( - 2, - 1)} \cr {\max \{ 2x,3[|x|]\} } & {,\,|x| where [t] d... If$$\alpha = \mathop {\lim }\limits_{x \to {\pi \over 4}} {{{{\tan }^3}x - \tan x} \over {\cos \left( {x + {\pi \over 4}} \right)}}$$and$$\beta ...
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
The function $$f(x) = \left| {{x^2} - 2x - 3} \right|\,.\,{e^{\left| {9{x^2} - 12x + 4} \right|}}$$ is not differentiable at exactly :
If the function $$f(x) = \left\{ {\matrix{ {{1 \over x}{{\log }_e}\left( {{{1 + {x \over a}} \over {1 - {x \over b}}}} \right)} & , & {x &l...$$\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}\left( {\pi {{\cos }^4}x} \right)} \over {{x^4}}}$$is equal to : If$$\mathop {\lim }\limits_{x \to \infty } \left( {\sqrt {{x^2} - x + 1} - ax} \right) = b$$, then the ordered pair (a, b) is : If$$\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^...
$$\mathop {\lim }\limits_{x \to 2} \left( {\sum\limits_{n = 1}^9 {{x \over {n(n + 1){x^2} + 2(2n + 1)x + 4}}} } \right)$$ is equal to :
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 :...
Let $$f:[0,\infty ) \to [0,3]$$ be a function defined by $$f(x) = \left\{ {\matrix{ {\max \{ \sin t:0 \le t \le x\} ,} & {0 \le x \le \pi } \c... Let$$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...
Let f : R $$\to$$ R be a function such that f(2) = 4 and f'(2) = 1. Then, the value of $$\mathop {\lim }\limits_{x \to 2} {{{x^2}f(2) - 4f(x)} \over {... Let$$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]$$. Then, f is : Let f : R$$\to$$R be defined as$$f(x) = \left\{ {\matrix{ {{{\lambda \left| {{x^2} - 5x + 6} \right|} \over {\mu (5x - {x^2} - 6)}},} & {x &l...
Let f : R $$\to$$ R be defined as$$f(x) = \left\{ {\matrix{ { - {4 \over 3}{x^3} + 2{x^2} + 3x,} & {x > 0} \cr {3x{e^x},} & {x \le ... Let f : R$$\to$$R be defined as$$f(x) = \left\{ {\matrix{ {{{{x^3}} \over {{{(1 - \cos 2x)}^2}}}{{\log }_e}\left( {{{1 + 2x{e^{ - 2x}}} \over {{...
If $$f:R \to R$$ is given by $$f(x) = x + 1$$, then the value of $$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\left[ {f(0) + f\left( {{5 \over ... The sum of all the local minimum values of the twice differentiable function f : R$$\to$$R defined by$$f(x) = {x^3} - 3{x^2} - {{3f''(2)} \over 2}x...
Let a function f : R $$\to$$ R be defined as $$f(x) = \left\{ {\matrix{ {\sin x - {e^x}} & {if} & {x \le 0} \cr {a + [ - x]} & {if... Let f : R$$ \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\...
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
If $$f(x) = \left\{ {\matrix{ {{1 \over {|x|}}} & {;\,|x|\, \ge 1} \cr {a{x^2} + b} & {;\,|x|\, < 1} \cr } } \right.$$ is diffe...
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...
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...
The value of $$\mathop {\lim }\limits_{x \to {0^ + }} {{{{\cos }^{ - 1}}(x - {{[x]}^2}).{{\sin }^{ - 1}}(x - {{[x]}^2})} \over {x - {x^3}}}$$, where [...
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...
Let $$\alpha$$ $$\in$$ R be such that the function $$f(x) = \left\{ {\matrix{ {{{{{\cos }^{ - 1}}(1 - {{\{ x\} }^2}){{\sin }^{ - 1}}(1 - \{ x\} )} ... Let$${S_k} = \sum\limits_{r = 1}^k {{{\tan }^{ - 1}}\left( {{{{6^r}} \over {{2^{2r + 1}} + {3^{2r + 1}}}}} \right)} $$. Then$$\mathop {\lim }\limits...
Let the functions f : R $$\to$$ R and g : R $$\to$$ R be defined as :$$f(x) = \left\{ {\matrix{ {x + 2,} & {x < 0} \cr {{x^2},} &am... Let f(x) be a differentiable function at x = a with f'(a) = 2 and f(a) = 4. Then$$\mathop {\lim }\limits_{x \to a} {{xf(a) - af(x)} \over {x - a}}$$... Let$$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)$$, then the ... Let f : R$$ \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 ...
The value of $$\mathop {\lim }\limits_{h \to 0} 2\left\{ {{{\sqrt 3 \sin \left( {{\pi \over 6} + h} \right) - \cos \left( {{\pi \over 6} + h} \right... Let f be any function defined on R and let it satisfy the condition :$$|f(x) - f(y)|\, \le \,|{(x - y)^2}|,\forall (x,y) \in R$$If f(0) = 1, then :...$$\mathop {\lim }\limits_{n \to \infty } {\left( {1 + {{1 + {1 \over 2} + ........ + {1 \over n}} \over {{n^2}}}} \right)^n}$$is equal to : If f : R$$ \to $$R is a function defined by f(x)= [x - 1]$$\cos \left( {{{2x - 1} \over 2}} \right)\pi $$, where [.] denotes the greatest integer f... Let f : R$$ \to $$R be a function defined by f(x) = max {x, x2}. Let S denote the set of all points in R, where f is not differentiable. Then :... For all twice differentiable functions f : R$$ \to $$R, with f(0) = f(1) = f'(0) = 0$$\mathop {\lim }\limits_{x \to 1} \left( {{{\int\limits_0^{{{\left( {x - 1} \right)}^2}} {t\cos \left( {{t^2}} \right)dt} } \over {\left( {x - 1} \ri...
$$\mathop {\lim }\limits_{x \to 0} {{x\left( {{e^{\left( {\sqrt {1 + {x^2} + {x^4}} - 1} \right)/x}} - 1} \right)} \over {\sqrt {1 + {x^2} + {x^4}} ... If$$\alpha $$is positive root of the equation, p(x) = x2 - x - 2 = 0, then$$\mathop {\lim }\limits_{x \to {\alpha ^ + }} {{\sqrt {1 - \cos \left( {...
Let $$f:\left( {0,\infty } \right) \to \left( {0,\infty } \right)$$ be a differentiable function such that f(1) = e and $$\mathop {\lim }\limits_{t \t... The function$$f(x) = \left\{ {\matrix{ {{\pi \over 4} + {{\tan }^{ - 1}}x,} & {\left| x \right| \le 1} \cr {{1 \over 2}\left( {\left| x ...
$$\mathop {\lim }\limits_{x \to a} {{{{\left( {a + 2x} \right)}^{{1 \over 3}}} - {{\left( {3x} \right)}^{{1 \over 3}}}} \over {{{\left( {3a + x} \righ... Let [t] denote the greatest integer$$ \le $$t. If for some$$\lambda  \in $$R - {1, 0},$$\mathop {\lim }\limits_{x \to 0} \left| {{{1 - x + \...
$$\mathop {\lim }\limits_{x \to 0} {\left( {\tan \left( {{\pi \over 4} + x} \right)} \right)^{{1 \over x}}}$$ is equal to :
If a function f(x) defined by $$f\left( x \right) = \left\{ {\matrix{ {a{e^x} + b{e^{ - x}},} & { - 1 \le x < 1} \cr {c{x^2},} & {1... Let [t] denote the greatest integer$$ \le $$t and$$\mathop {\lim }\limits_{x \to 0} x\left[ {{4 \over x}} \right] = A$$. Then the function, f(x) = ... Let a function ƒ : [0, 5]$$ \to $$R be continuous, ƒ(1) = 3 and F be defined as :$$F(x) = \int\limits_1^x {{t^2}g(t)dt} $$, where$$g(t) = \int\li...
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...
If $$f(x) = \left\{ {\matrix{ {{{\sin (a + 2)x + \sin x} \over x};} & {x < 0} \cr {b\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\... Let S be the set of all functions ƒ : [0,1]$$ \to $$R, which are continuous on [0,1] and differentiable on (0,1). Then for every ƒ in S, there exist...$$\mathop {\lim }\limits_{x \to 0} {{\int_0^x {t\sin \left( {10t} \right)dt} } \over x}$$is equal to$$\mathop {\lim }\limits_{x \to 0} {\left( {{{3{x^2} + 2} \over {7{x^2} + 2}}} \right)^{{1 \over {{x^2}}}}}$$is equal to The value of c in the Lagrange's mean value theorem for the function ƒ(x) = x3 - 4x2 + 8x + 11, when x$$ \in $$[0, 1] is: ... Let ƒ(x) be a polynomial of degree 5 such that x = ±1 are its critical points. If$$\mathop {\lim }\limits_{x \to 0} \left( {2 + {{f\left( x \right)} ...
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) $$\le$$ 2, for all x $$\in$$...
$$\mathop {\lim }\limits_{x \to 0} {{x + 2\sin x} \over {\sqrt {{x^2} + 2\sin x + 1} - \sqrt {{{\sin }^2}x - x + 1} }}$$ is :
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...
If $$\alpha$$ and $$\beta$$ are the roots of the equation 375x2 – 25x – 2 = 0, then $$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {... If$$\mathop {\lim }\limits_{x \to 1} {{{x^2} - ax + b} \over {x - 1}} = 5$$, then a + b is equal to : If$$\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}}}$$, then ... If$$f(x) = \left\{ {\matrix{ {{{\sin (p + 1)x + \sin x} \over x}} & {,x < 0} \cr q & {,x = 0} \cr {{{\sqrt {x + {x^2}} - \sqr...
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 :
$$\mathop {\lim }\limits_{n \to \infty } \left( {{{{{(n + 1)}^{1/3}}} \over {{n^{4/3}}}} + {{{{(n + 2)}^{1/3}}} \over {{n^{4/3}}}} + ....... + {{{{(2n... If$$f(x) = [x] - \left[ {{x \over 4}} \right]$$,x$$ \in $$4 , where [x] denotes the greatest integer function, then If the function$$f(x) = \left\{ {\matrix{ {a|\pi - x| + 1,x \le 5} \cr {b|x - \pi | + 3,x > 5} \cr } } \right.$$is continuous at x =... Let ƒ(x) = 15 – |x – 10|; x$$ \in $$R. Then the set of all values of x, at which the function, g(x) = ƒ(ƒ(x)) is not differentiable, is : If the function ƒ defined on ,$$\left( {{\pi \over 6},{\pi \over 3}} \right)$$by$$\$f(x) = \left\{ {\matrix{ {{{\sqrt 2 {\mathop{\rm cosx}\noli...
Let ƒ : R $$\to$$ R be a differentiable function satisfying ƒ'(3) + ƒ'(2) = 0. Then $$\mathop {\lim }\limits_{x \to 0} {\left( {{{1 + f(3 + x) - f(3... Let ƒ : [–1,3]$$ \to $$R be defined as$$f(x) = \left\{ {\matrix{ {\left| x \right| + \left[ x \right]} & , & { - 1 \le x < 1} \cr ...
$$\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}x} \over {\sqrt 2 - \sqrt {1 + \cos x} }}$$ equals:
$$\mathop {\lim }\limits_{x \to {1^ - }} {{\sqrt \pi - \sqrt {2{{\sin }^{ - 1}}x} } \over {\sqrt {1 - x} }}$$ is equal to :
$$\mathop {\lim }\limits_{x \to \infty } \left( {{n \over {{n^2} + {1^2}}} + {n \over {{n^2} + {2^2}}} + {n \over {{n^2} + {3^2}}} + ..... + {1 \over ... Let f be a differentiable function such that f(1) = 2 and f '(x) = f(x) for all x$$ \in $$R R. If h(x) = f(f(x)), then h'(1) is equal to : Let f and g be continuous functions on [0, a] such that f(x) = f(a – x) and g(x) + g(a – x) = 4, then$$\int\limits_0^a \, $$f(x) g(x) dx is equal to ...$$\mathop {\lim }\limits_{x \to \pi /4} {{{{\cot }^3}x - \tan x} \over {\cos \left( {x + {\pi \over 4}} \right)}}$$is : Let S be the set of all points in (–$$\pi $$,$$\pi $$) at which the function, f(x) = min{sin x, cos x} is not differentiable. Then S is a subset of w... Let K be the set of all real values of x where the function f(x) = sin |x| – |x| + 2(x –$$\pi $$) cos |x| is not differentiable. Then the set K is eq...$$\mathop {\lim }\limits_{x \to 0} {{x\cot \left( {4x} \right)} \over {{{\sin }^2}x{{\cot }^2}\left( {2x} \right)}}$$is equal to : Let$$f\left( x \right) = \left\{ {\matrix{ { - 1} & { - 2 \le x < 0} \cr {{x^2} - 1,} & {0 \le x \le 2} \cr } } \right.$$and... Let [x] denote the greatest integer less than or equal to x. Then$$\mathop {\lim }\limits_{x \to 0} {{\tan \left( {\pi {{\sin }^2}x} \right) + {{\lef...
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 ...
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 -
For each t $$\in$$ R , let [t] be the greatest integer less than or equal to t Then  $$\mathop {\lim }\limits_{x \to 1 + } {{\left( {1 - \... Let$$f\left( x \right) = \left\{ {\matrix{ {\max \left\{ {\left| x \right|,{x^2}} \right\}} & {\left| x \right| \le 2} \cr {8 ...
For each x$$\in$$R, let [x] be the greatest integer less than or equal to x. Then $$\mathop {\lim }\limits_{x \to {0^ - }} \,\,{{x\left( {\left[ x ... Let f be a differentiable function from R to R such that$$\left| {f\left( x \right) - f\left( y \right)} \right| \le 2{\left| {x - y} \right|^{{3 \o...
Let f : R $$\to$$ R be a function defined as $$f(x) = \left\{ {\matrix{ 5 & ; & {x \le 1} \cr {a + bx} & ; & {1 < x <...$$\mathop {\lim }\limits_{y \to 0} {{\sqrt {1 + \sqrt {1 + {y^4}} } - \sqrt 2 } \over {{y^4}}}\mathop {\lim }\limits_{x \to 0} \,\,{{{{\left( {27 + x} \right)}^{{1 \over 3}}} - 3} \over {9 - {{\left( {27 + x} \right)}^{{2 \over 3}}}}}$$equal... If$$x = \sqrt {{2^{\cos e{c^{ - 1}}}}} $$and$$y = \sqrt {{2^{se{c^{ - 1}}t}}} \,\,\left( {\left| t \right| \ge 1} \right),$$then$${{dy} \over {dx...
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 ...
For each t $$\in R$$, let [t] be the greatest integer less than or equal to t. Then $$\mathop {\lim }\limits_{x \to {0^ + }} x\left( {\left[ {{1 \ove...$$\mathop {\lim }\limits_{x \to 0} {{x\tan 2x - 2x\tan x} \over {{{\left( {1 - \cos 2x} \right)}^2}}}$$equals : Let f(x) =$$\left\{ {\matrix{ {{{\left( {x - 1} \right)}^{{1 \over {2 - x}}}},} & {x > 1,x \ne 2} \cr {k\,\,\,\,\,\,\,\,\,\,\,\,\,\,} ...
If    f(x) = sin-1 $$\left( {{{2 \times {3^x}} \over {1 + {9^x}}}} \right),$$ then f'$$\left( { - {1 \over 2}} \right)$$ equals :
Let f(x) be a polynomial of degree $$4$$ having extreme values at $$x = 1$$ and $$x = 2.$$ If   $$\mathop {lim}\limits_{x \to 0} \left( {{{f\left... If$$f\left( x \right) = \left| {\matrix{ {\cos x} & x & 1 \cr {2\sin x} & {{x^2}} & {2x} \cr {\tan x} & x & 1 \...
If   x2 + y2 + sin y = 4, then the value of $${{{d^2}y} \over {d{x^2}}}$$ at the point ($$-$$2,0) is :
Let S = {($$\lambda$$, $$\mu$$) $$\in$$ R $$\times$$ R : f(t) = (|$$\lambda$$| e|t| $$-$$ $$\mu$$). sin (2|t|), t $$\in$$ R, is a differen...
The value of k for which the function $$f\left( x \right) = \left\{ {\matrix{ {{{\left( {{4 \over 5}} \right)}^{{{\tan \,4x} \over {\tan \,5x}}}}\...$$\mathop {\lim }\limits_{x \to 3} {{\sqrt {3x} - 3} \over {\sqrt {2x - 4} - \sqrt 2 }}$$is equal to :$$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{\cot x - \cos x} \over {{{\left( {\pi - 2x} \right)}^3}}}$$equals If for$$x \in \left( {0,{1 \over 4}} \right)$$, the derivatives of$${\tan ^{ - 1}}\left( {{{6x\sqrt x } \over {1 - 9{x^3}}}} \right)$$is$$\sqrt x ...
$$\mathop {\lim }\limits_{x \to 0} \,{{{{\left( {1 - \cos 2x} \right)}^2}} \over {2x\,\tan x\, - x\tan 2x}}$$ is :
Let a, b $$\in$$ R, (a $$\ne$$ 0). If the function f defined as $$f\left( x \right) = \left\{ {\matrix{ {{{2{x^2}} \over a}\,\,,} & {0 \le ... If$$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} - {4 \over {{x^2}}}} \right)^{2x}} = {e^3},$$then 'a' is equal to :... If the function f(x) =$$\left\{ {\matrix{ { - x} & {x < 1} \cr {a + {{\cos }^{ - 1}}\left( {x + b} \right),} & {1 \le x \le 2} \...
$$\mathop {\lim }\limits_{n \to \infty } {\left( {{{\left( {n + 1} \right)\left( {n + 2} \right)...3n} \over {{n^{2n}}}}} \right)^{{1 \over n}}}$$ is ...
For $$x \in \,R,\,\,f\left( x \right) = \left| {\log 2 - \sin x} \right|\,\,$$ and $$\,\,g\left( x \right) = f\left( {f\left( x \right)} \right),\,\,... Let$$p = \mathop {\lim }\limits_{x \to {0^ + }} {\left( {1 + {{\tan }^2}\sqrt x } \right)^{{1 \over {2x}}}}$$then$$logp$$is equal to :$$\mathop {\lim }\limits_{x \to 0} {{\left( {1 - \cos 2x} \right)\left( {3 + \cos x} \right)} \over {x\tan 4x}}$$is equal to If the function.$$g\left( x \right) = \left\{ {\matrix{ {k\sqrt {x + 1} ,} & {0 \le x \le 3} \cr {m\,x + 2,} & {3 < x \le 5} \cr ...
$$\mathop {\lim }\limits_{x \to 0} {{\sin \left( {\pi {{\cos }^2}x} \right)} \over {{x^2}}}$$ is equal to :
$$\mathop {\lim }\limits_{x \to 0} {{\left( {1 - \cos 2x} \right)\left( {3 + \cos x} \right)} \over {x\tan 4x}}$$ is equal to
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...
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 ...
$$\mathop {\lim }\limits_{x \to 2} \left( {{{\sqrt {1 - \cos \left\{ {2(x - 2)} \right\}} } \over {x - 2}}} \right)$$
The value of $$p$$ and $$q$$ for which the function $$f\left( x \right) = \left\{ {\matrix{ {{{\sin (p + 1)x + \sin x} \over x}} & {,x < 0} ... Let$$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 }...
Let $$f\left( x \right) = \left\{ {\matrix{ {\left( {x - 1} \right)\sin {1 \over {x - 1}}} & {if\,x \ne 1} \cr 0 & {if\,x = 1} \cr ... Let$$f:R \to R$$be a function defined by$$f(x) = \min \left\{ {x + 1,\left| x \right| + 1} \right\}$$, then which of the following is true? The function$$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$$...$$\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over {{n^2}}}{{\sec }^2}{1 \over {{n^2}}} + {2 \over {{n^2}}}{{\sec }^2}{4 \over {{n^2}}}.... + {...
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 $$f'\left( 1 \right)... Let$$\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}...
Let f be differentiable for all x. If f(1) = -2 and f'(x) $$\ge$$ 2 for x $$\in \left[ {1,6} \right]$$, then
If $$f$$ is a real valued differentiable function satisfying $$\left| {f\left( x \right) - f\left( y \right)} \right|$$ $$\le {\left( {x - y} \right)... Let$$f(x) = {{1 - \tan x} \over {4x - \pi }}$$,$$x \ne {\pi \over 4}$$,$$x \in \left[ {0,{\pi \over 2}} \right]$$. If$$f(x)$$is continuous in ... If$$\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$$, ...$$\mathop {\lim }\limits_{n \to \infty } {{1 + {2^4} + {3^4} + .... + {n^4}} \over {{n^5}}}$$-$$\mathop {\lim }\limits_{n \to \infty } {{1 + {2^3} +...
The value of $$\mathop {\lim }\limits_{x \to 0} {{\int\limits_0^{{x^2}} {{{\sec }^2}tdt} } \over xsinx}$$ is
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
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 $$\mathop {\lim }\limits_...$$\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 ...
If $$f(x) = \left\{ {\matrix{ {x{e^{ - \left( {{1 \over {\left| x \right|}} + {1 \over x}} \right)}}} & {,x \ne 0} \cr 0 & {,x = 0} \...$$\mathop {\lim }\limits_{x \to 0} {{\sqrt {1 - \cos 2x} } \over {\sqrt 2 x}}$$is Let$$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}}$$is given by...$$\mathop {\lim }\limits_{n \to \infty } {{{1^p} + {2^p} + {3^p} + ..... + {n^p}} \over {{n^{p + 1}}}}$$is$$\mathop {\lim }\limits_{x \to \infty } {\left( {{{{x^2} + 5x + 3} \over {{x^2} + x + 2}}} \right)^x}\mathop {\lim }\limits_{x \to 0} {{\log {x^n} - \left[ x \right]} \over {\left[ x \right]}}$$,$$n \in N$$, ( [x] denotes the greatest integer less ... If$$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}...
$$f$$ is defined in $$\left[ { - 5,5} \right]$$ as $$f\left( x \right) = x$$ if $$x$$ is rational $$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$= - x$$ if $$x$$ ...
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...
If f(x + y) = f(x).f(y) $$\forall$$ x, y and f(5) = 2, f'(0) = 3, then f'(5) is

## Numerical

If $$[t]$$ denotes the greatest integer $$\leq t$$, then the number of points, at which the function $$f(x)=4|2 x+3|+9\left[x+\frac{1}{2}\right]-12[x+...$$\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...
Suppose $$\mathop {\lim }\limits_{x \to 0} {{F(x)} \over {{x^3}}}$$ exists and is equal to L, where $$F(x) = \left| {\matrix{ {a + \sin {x \over 2}... If$$\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$$, then the value of (a$$-$$b) ... Let$$f(x) = \left[ {2{x^2} + 1} \right]$$and$$g(x) = \left\{ {\matrix{ {2x - 3,} & {x ...
Let $$f(x) = {x^6} + 2{x^4} + {x^3} + 2x + 3$$, x $$\in$$ R. Then the natural number n for which $$\mathop {\lim }\limits_{x \to 1} {{{x^n}f(1) - f(x)... Let [t] denote the greatest integer$$\le$$t. The number of points where the function$$f(x) = [x]\left| {{x^2} - 1} \right| + \sin \left( {{\pi \ov...
Let a, b $$\in$$ R, b $$\in$$ 0, Define a function $$f(x) = \left\{ {\matrix{ {a\sin {\pi \over 2}(x - 1),} & {for\,x \le 0} \cr {{{\tan ... Let$$f:[0,3] \to R$$be defined by$$f(x) = \min \{ x - [x],1 + [x] - x\} $$where [x] is the greatest integer less than or equal to x. Let P denote ... Consider the functionwhere P(x) is a polynomial such that P'' (x) is always a constant and P(3) = 9. If f(x) is continuous at x = 2, then P(5) is equa... Let f : R$$\to$$R be a function defined as$$f(x) = \left\{ {\matrix{ {3\left( {1 - {{|x|} \over 2}} \right)} & {if} & {|x|\, \le 2} \cr...
Let a function g : [ 0, 4 ] $$\to$$ R be defined as $$g(x) = \left\{ {\matrix{ {\mathop {\max }\limits_{0 \le t \le x} \{ {t^3} - 6{t^2} + 9t - 3),... If$$\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 ,...
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 ...
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...
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 __...
Let f : R $$\to$$ R and g : R $$\to$$ R be defined as $$f(x) = \left\{ {\matrix{ {x + a,} & {x < 0} \cr {|x - 1|,} & {x \ge 0} ... Let f : (0, 2)$$ \to $$R be defined as f(x) = log2$$\left( {1 + \tan \left( {{{\pi x} \over 4}} \right)} \right)$$. Then,$$\mathop {\lim }\limits_{...
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 ____________.
A function f is defined on [$$-$$3, 3] as$$f(x) = \left\{ {\matrix{ {\min \{ |x|,2 - {x^2}\} ,} & { - 2 \le x \le 2} \cr {[|x|],} & {2... If$$\mathop {\lim }\limits_{x \to 0} {{ax - ({e^{4x}} - 1)} \over {ax({e^{4x}} - 1)}}$$exists and is equal to b, then the value of a$$-$$2b is ___... The number of points, at which the function f(x) = | 2x + 1 |$$-$$3| x + 2 | + | x2 + x$$-$$2 |, x$$\in$$R is not differentiable, is __________.... Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x6 is unity and it has extrema at x =$$-$$1 and x = 1. If$$\mathop {\lim }\li...
$$\mathop {\lim }\limits_{n \to \infty } \tan \left\{ {\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {{1 \over {1 + r + {r^2}}}} \right)} } \right\}$$...
Let f : R $$\to$$ R be defined as $$f\left( x \right) = \left\{ {\matrix{ {{x^5}\sin \left( {{1 \over x}} \right) + 5{x^2},} & {x < 0} \c... Let$$f(x) = x.\left[ {{x \over 2}} \right]$$, for -10< x < 10, where [t] denotes the greatest integer function. Then the number of points of d... Suppose a differentiable function f(x) satisfies the identity f(x+y) = f(x) + f(y) + xy2 + x2y, for all real x and y.$$\mathop {\lim }\limits_{x \to ...
If $$\mathop {\lim }\limits_{x \to 0} \left\{ {{1 \over {{x^8}}}\left( {1 - \cos {{{x^2}} \over 2} - \cos {{{x^2}} \over 4} + \cos {{{x^2}} \over 2}\c... If$$\mathop {\lim }\limits_{x \to 1} {{x + {x^2} + {x^3} + ... + {x^n} - n} \over {x - 1}}$$= 820, (n$$ \in $$N) then the value of n is equal to _... If the function ƒ defined on$$\left( { - {1 \over 3},{1 \over 3}} \right)$$by f(x) =$$\left\{ {\matrix{ {{1 \over x}{{\log }_e}\left( {{{1 + 3x}...
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...
$$\mathop {\lim }\limits_{x \to 2} {{{3^x} + {3^{3 - x}} - 12} \over {{3^{ - x/2}} - {3^{1 - x}}}}$$ is equal to_______.
EXAM MAP
Joint Entrance Examination