Limits Questions - JEE

Let $f(x)$ be a polynomial of the second degree which is positive definite. If $g(x) = f(x) + f'(x) + f''(x)$ then for any real $x$

If $y = x + e^x$, then $\dfrac{d^2x}{dy^2}$ is:

If $x = y\ln(xy)$, then $\dfrac{dx}{dy}$ equals:

Given $f(1) = 5$, $f(2) = 8$, $f'(1) = 3$ and $f''(1) = 0$, find $f(3)$.

Evaluate: $\lim_{x \to 0^+} \dfrac{\displaystyle\int_0^{\tan^{-1} x} (\sin t^2)\, dt}{\dfrac{-x^3}{2}}$

Evaluate: $\lim_{x \to 0} \dfrac{1}{x} - \dfrac{2}{e^{2x} - 1}$

Find \lim_{x \to 0^-} f(x) and \lim_{x \to 1^-} f(x), where f(x) = \begin{cases} 2x + 3, & x \leq 0 \\ 3(x+1), & x > 0 \end{cases}

If $ f $ and $ g $ are differentiable functions in $[0, 1]$ satisfying $ f(0) = 2 = g(1) $, $ g(0) = 0 $ and $ f(1) = 6 $, then for some $ c \in [0, 1] $

The number of discontinuity of the greatest integer function $f(x) = [x]$, $x \in (-\frac{1}{2}, 100)$ is equal to

Let f(x) be a continuous function satisfying $\begin{cases} f(x) = x^2 - 6x + 8 & \text{for } -1 \le x \le a-1 \\ f(x) = f(x+a) \end{cases}$ where 'a' is a constant, then find the sum of all possible values of 'a'.

Let $f(x)$ be a continuous, periodic and bounded function with period 3 such that $\int_0^3 f(t)\,dt = 6$. Also $g'(x) = f(x)$, such that $g(0) = 0$. Find the value of $\lim_{x \to 0} x g\!\left(\dfrac{1}{x}\right)$.

Let $ f(x) = x|x| $ and $ g(x) = \sin x $.Statement-1: $ g \circ f $ is differentiable at $ x = 0 $ and its derivative is continuous at that point.Statement-2: $ g \circ f $ is twice differentiable at $ x = 0 $.

If $\lim_{x \to 1} \dfrac{x^4 - 1}{x - 1} = \lim_{x \to k} \dfrac{x^3 - k^3}{x^2 - k^2}$, then $k$ is __________ (up to two decimal places).

$\lim_{x \to \infty} \left(\dfrac{x^2 + 5x + 3}{x^2 + x + 2}\right)^x$ is equal to

If $f(x) = \sqrt{x - 4\sqrt{x - 4}} + \tan^{-1}\left(\frac{1+x}{1-x}\right), 4 < x < 8$, then the value of $f'(5)$ is equal to

$\lim_{x \to 0} \dfrac{(1 - \cos 2x)^2}{2x \tan x - x \tan 2x}$ is

If $f(x) = |x|^{\{x\}} + (\{x\})^2 + \sin(\pi x)$, where $[\cdot]$ and $\{\cdot\}$ represent the greatest integer function and the fractional part function respectively, then $f'\left(\frac{3}{2}\right)$ is

Let $f:(-2,2)\to\mathbb{R}$ be defined by $f(x)=\begin{cases}x[x], & -2<x<0\\ (x-1)[x], & 0\leq x<2\end{cases}$ where $[x]$ denotes the greatest integer function. If $m$ and $n$ respectively are the number of points in $(-2,2)$ at which $y=|f(x)|$ is not continuous and not differentiable, then $m+n$ is equal to ________.

Let $f(x)$ be a differentiable function on $x \in R$, such that $f(x + y) = f(x)f(y)$ for all $x, y \in R$ where $f(0) \neq 0$. If $f'(5) = 10$, $f'(0) = 0$, then the value of $f'(5)$ is equal to

Given that $ f(x) = \dfrac{\sqrt{2}\cos x - 1}{\cot x - 1} $ is continuous at $ x = \dfrac{\pi}{4} $. Find the value of $ f\!\left(\dfrac{\pi}{4}\right) = k $.

If $x^{2x} - 2x^x \cot y - 1 = 0$, find $y'(1)$.

In $(0, 2\pi)$, the total number of points where $f(x) = \max(\sin x, \cos x, 1 - \cos x)$ is not differentiable, are equal to

The number of points at which the function $f(x) = |x - 0.5| + |x - 1| + \tan x$ is not differentiable in the interval $(0, 2)$ is/are

For $a,b>0$, let $f(x)=\begin{cases}\dfrac{\tan((a+1)x)+b\tan x}{x}, & x<0\\ 3, & x=0\\ \dfrac{\sqrt{ax+b^2x^2}-\sqrt{ax}}{b\sqrt{ax}\sqrt{x}}, & x>0\end{cases}$ be a continuous function at $x=0$. Then $\dfrac{b}{a}$ is equal to:

$\lim_{x \to 0} \dfrac{\sin^2 x}{\sqrt{2} - \sqrt{1 + \cos x}}$ equals __________ (up to four decimal places).

Consider the function $f(x) = \max\{\sin x, |\cos x|\}, \forall x \in [0, 2\pi]$. If $n$ is the number of points at which $f(x)$ is non-differentiable, then the value of $n$ is

If $f(x) = \begin{cases} \frac{x \ln(e + \cos x)}{x^2} & x > 0 \\ q & x \leq 0 \end{cases}$ is continuous at $x = 0$, then the value of $\frac{pq+1}{q}$ is

If $y = \tan^{-1}\left(\frac{1}{x^2+1}\right) + \tan^{-1}\left(\frac{1}{x^2+3}\right)$, find $\frac{dy}{dx}$

If $f(n+1) = \dfrac{1}{2}\left\{f(n) + \dfrac{9}{f(n)}\right\}$ where $n \in N$ and $f(n) > 0\ \forall n \in N$ and $\underset{n\to\infty}{\text{Lt}}\ f(n)$ exist then the value of $\underset{n\to\infty}{\text{Lt}}\ f(n) =$

If $f(x)$ is a real valued bijective function satisfying $f'(x) = \sin^2(\sin(x+1))$ and $f(0) = 3$, then the value of $(f^{-1})''(3)$ is equal to:

Let f be a real valued derivable function such that f(x)f(y) = f(x)y + xf(y), ∀x, y ∈ ℝ. If f′(0) = 2, then find $\lim_{x \to 0} \left[\dfrac{f(x)}{\sin x}\right]$. [Note: [ ] represents greatest integer function.]

Let $f$ and $g$ be two functions defined by $f(x)=\begin{cases}x+1, & x<0\\|x-1|, & x\geq0\end{cases}$ and $g(x)=\begin{cases}x+1, & x<0\\1, & x\geq0\end{cases}$. Then $(g\circ f)(x)$ is

Let $f(x)=[x^2-x]+|-x+[x]|$, where $x\in\mathbb{R}$ and $[t]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is

Let $ f(x) = \dfrac{1 - \tan x}{4x - \pi} $, $ x \neq \dfrac{\pi}{4} $, $ x \in \left[0, \dfrac{\pi}{2}\right] $. If $ f(x) $ is continuous in $ \left[0, \dfrac{\pi}{2}\right] $, then $ f\!\left(\dfrac{\pi}{4}\right) $ is

If lim x\to\infty (( e )( 1 - x )) = \alpha , then the value of log e \alpha equals : 1-e e 1+x 1+log \alpha e

If the function 2 ⎧ {sin(k1 + 1)x + sin(k2 - 1)x} , x < 0 ⎪ ⎪ x ⎪ 4, x = 0 f (x) = ⎨ ⎪ ⎪ 2 2+k1 x ⎩ ⎪ log ( ), x > 0 x e 2+k2 x is continuous at x = 0, then k + k is equal to 2 1 2 2

$\displaystyle\lim_{n\to\infty}\left\{\left(2^{1/2}-2^{1/3}\right)\left(2^{1/2}-2^{1/5}\right)\cdots\left(2^{1/2}-2^{1/(2n+1)}\right)\right\}$ is equal to

Evaluate $\lim_{x \to \frac{\pi}{2}} \frac{\tan\left(\frac{x}{2}\right)(1-\sin x)}{(x-2y)^3}$

We have \[\lim_{x \to 1} \left(1 + \frac{a}{x} - \frac{4}{x^2}\right)^{2x} = e^3\] Find the value of $a$.

If $y = e^{ax^{-4x}}$ and $z = e^{-\ cos^{-1}x}$ then $\dfrac{d^2y}{dz^2} = 0$.State whether the statement is true or false.

Find: $\lim_{n \to \infty} \left(\frac{1^{1/x} + 2^{1/x} + 3^{1/x} + \cdots + n^{1/x}}{n}\right)^{nx}$

If $f(1) = g(1) = 2$ and $f'(1)$, $g'(1)$ exist, then evaluate \[\lim_{x \to 1} \frac{f(1)g(x) - f(x)g(1)}{g(x) - f(x)}\]

$\lim_{x \to 0} \frac{2e^{\sin x} - e^{-\sin x} - 1}{x^2 + 2x}$ equals:

If $g(x) = (x^2 + 2x + 3)f(x)$, $f(0) = 5$ and $\displaystyle\lim_{x \to 0}\left(\frac{f(x) - 5}{x}\right) = 4$, then $g'(0)$ is equal to:

The value of $\lim_{x \to 0} \frac{\sin x + \cos(e^x)}{1 + \sin(3x)}$ is equal to

Let $f(x) = 10 - |x - 5|, x \in \mathbb{R}$, then the set of all values of $x$ at which $f(f(x))$ is not differentiable is

Let $f(x) = \sin(\cos^{-1}(\sin x))$ and $g(x) = \cos(\sin^{-1}(\cos x))$. If $S$ is the range of $\dfrac{f'(x)}{g'(x)}$, then $S$ contains:

If $\displaystyle\lim_{x\to0}\dfrac{e^{ax}-\cos(bx)-cxe^{-cx}}{1-\cos(2x)}=17$, then $5a^2+b^2$ is equal to

$\displaystyle\lim_{x\to0}\frac{e-(1+2x)^{\frac{1}{2x}}}{x}$ is equal to

Let $a\in\mathbb{Z}$ and $[t]$ be the greatest integer $\leq t$, then the number of points, where the function $f(x)=[a+13\sin x]$, $x\in(0,\pi)$ is not differentiable, is ____________.

Limits Questions (1092)