Limits, Continuity & Differentiability Questions - JEE

Given $ x = 3\tan t,\ y = 3\sec t $, find $\left(\dfrac{d^2 y}{dx^2}\right)_{t=\pi/4}$.

If $\displaystyle\lim_{\alpha \to 0} \frac{e^{\cos(\alpha^n)} - e}{\alpha^m} = \frac{-e}{2}$ where $m$ and $n$ are positive integers greater than 1, then the value of $\dfrac{m}{n}$ is:

$\lim_{x \to \frac{\pi}{2}} \dfrac{\cot x - \cos x}{(\pi - 2x)^3}$ equals

The value of $\displaystyle\lim_{x \to 0}\left\lfloor (1-e^x)\frac{\sin x}{|x|}\right\rfloor$ equals: [Note: $[\,\cdot\,]$ denotes the greatest integer function.]

$f(x) = x^2 + 3, x \leq 1$$= 3x + a, x > 1$Is $f(x)$ neither continuous nor differentiable at $x = 1$?

Let $x^3 - 2x^2y^2 + 5x + y - 5 = 0$ and at $x = 1$, $y = 1$. Then $\dfrac{dy}{dx}$ at $y = 1$ is

If $ x\log_e(\log_e x) - x^2 + y^2 = 4 $ $(y > 0)$, then $ \dfrac{dy}{dx} $ at $ x = e $ is equal to:

$\lim_{x \to 2} [x]$ exists where $[x]$ denotes the integral part of $x$.State whether the statement is true or false.

If $f(x)$ is odd linear polynomial with $f(1) = 1$, then \[\lim_{x \to 0} \frac{2^{f(\tan x)} - 2^{f(\sin x)}}{x^2 f(\sin x)}\] is

\[\lim_{n \to \infty} \frac{e^n}{\left(1 + \dfrac{1}{n}\right)^{n^2}}\] equals

If $f(x)$ is a polynomial of least degree such that $\lim_{x \to 0}\left(1 + \dfrac{f(x) + x^2}{x^2}\right)^{1/x} = e^2$, then $f(2)$ is

$\lim_{x \to 0} \dfrac{e^{x^2} - \cos x}{\sin^2 x}$ is equal to

Let \(f(x) = \begin{cases} \sqrt{x^2 - 1}, & x \leq \sqrt{10} \\ (\sqrt{10}x - 7), & \sqrt{10}

Let $f(2) = 4$ and $f'(2) = 4$. Then $\displaystyle\lim_{x \to 2} \frac{xf(2) - 2f(x)}{x - 2}$ is given by

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 $f: R \to R$ be a function such that $|f(x)| \leq x^2$, for all $x \in R$. Then at $x = 0$, $f(x)$ is

If $ x = e^{y + e^{y + \cdots}} $, $ x > 0 $, then $ \dfrac{dy}{dx} $ is:

Given $f(t) = (|\lambda|e^{|t|} - \mu)\sin(2|t|)$. If $f(t)$ is differentiable at $t = 0$, then the set $S$ of all possible values of $(\lambda, \mu)$ is a subset of:

The value of $\displaystyle\lim_{x \to \infty} \frac{e^x\left[\left(2^{x^n}\right)^{1/e^x} - \left(e^{x^n}\right)^{1/e^x}\right]}{x^n}$ where $n$ is positive integer, is:

Given $f(x) = (x-1)^{\frac{1}{2-x}},\; x > 1,\; x \neq 2$ and $f(2) = k$. If $f$ is continuous at $x = 2$, find $k$.

If $\lim f(x)$ and $\lim g(x)$ exist then $\lim [f(x) \cdot g(x)]$ exist.

Let $ f(x) = \begin{cases} (x-1)\sin\left(\dfrac{1}{x-1}\right), & \text{if } x \neq 1 \\ 0, & \text{if } x = 1 \end{cases} $. Then which one of the following is true?

If $\sin y = x\sin(a+y)$, then $\dfrac{dy}{dx}$ equals:

Let $x_0 = \tan^{-1}(2)$ and \[b = \lim_{x \to \tan^{-1}2} \frac{(\tan^2 x - a)(1 + \tan x)}{e^{(\tan x - 2)} - 1}\] For the existence of the limit, find $[a + b + x_0]$ where $[\cdot]$ denotes the greatest integer function.

It is given that $2x = y^{1/5} + y^{-1/5}$If $y = (x + \sqrt{x^2 - 1})^5$, then $(x^2 - 1)\dfrac{d^2y}{dx^2} + \lambda x \dfrac{dy}{dx} - 25y = 0$. Find $\lambda + k$ where $k = -25$.

The value of $\lim_{x \to 0^+} x^m (\log x)^n$, $m, n \in \mathbb{N}$ is

Given $f(x) = \begin{cases} x^2e^{-x} & 0 \leq x \leq 1 \\ a \sin(x+1) \cos(2x-2) + bx^3 & 1 < x \leq 2 \end{cases}$. If $f(x)$ is differentiable at $x = 1$, then the value of $|a - b|$ is

The value of $\lim_{x \to 0}(f(x) + g(x) + 3)^{1/x}$ equal to:

Let $f(x) = \left(\dfrac{4}{5}\right)^{\frac{\tan 4x}{\tan 5x}}$. If $\displaystyle\lim_{x \to \pi/2} f(x) = k + \dfrac{2}{5}$, find the value of $k$.

If $f(x) = \cos x\cdot\cos 2x\cdot\cos 4x\cdot\cos 8x\cdot\cos 16x$, then $f'\!\left(\dfrac{\pi}{4}\right)$ equals:

Let f(x) = x|x|, g(x) = sin x and h(x) = (g ∘ f)(x). Then

If $\lim_{x \to 0} f(x)$ and $\lim_{x \to 0} g(x)$ exist then $\lim_{x \to 0} g(x)$ exist. $\lim_{x \to 0} g(x)$ exist.

Let $f(x) = \sin\!\left(\sin^{-1}(2x)+2\tan^{-1}(2x)\right)$. If $3f'(0)=$ (integer), find it.

Let $f(x)=x$, $g(x)=|1-f(x)|$, $h(x)=2-g(x)$, $L(x)=h(|x|)+|h(x)|$. Find the number of points where $L(x)$ is non-differentiable.

If $ y = \sec(\tan^{-1} x) $, then $ \dfrac{dy}{dx} $ at $ x = 1 $ is equal to:

If $x\log_e(\log_e x) - x^2 + y^2 = 4$, then $\dfrac{dy}{dx}$ at $x = e$ is equal to:

If $f(x)$ is a differentiable function in the interval $(0, \infty)$ such that $f(1) = 1$ and $\lim_{t \to x} \dfrac{t^2 f(x) - x^2 f(t)}{t - x} = 1$, for each $x > 0$, then $f(3/2)$ is equal to

State whether the statement is true or false: $\lim_{y \to 0} \frac{\log(1+x)}{3^y - 1} = \frac{1}{\log_e 3}$

Let $f$ and $g$ be inverse functions of each other. If $f(1) = 3$ and $f'(1) = 4$, then $g'(3)$ equals:

If $\lim_{x \to 0} \frac{1-\cos 2x \sin 5x}{x^2 \sin 3x} = \frac{10}{3}$State whether this statement is true or false.

$\lim_{x \to 0} \dfrac{(2x)^4\left(\dfrac{1-\cos 2x}{4x^2}\right)^2}{2x\left(\tan x - \dfrac{\tan 2x}{2}\right)}$

Consider the function $ f(x) = |x-2| + |x-5|, \, x \in R $.Statement-1: $ f'(4) = 0 $.Statement-2: $ f $ is continuous in $[2, 5]$, differentiable in $(2, 5)$ and $ f(2) = f(5) $.

If $f(x) = \tan^2\left[x - \frac{\pi}{4}\right]$ where $[x]$ is the greatest integer function then:

If $|f(x)| \leq x^2$, then which of the following is true?

Let $ f: R \to R $ be a function defined by $ f(x) = \min\{x+1,\ |x|+1\} $. Then which of the following is true?

If $f(x) = \begin{cases} x\tan^{-1}x\cdot\sec^{-1}\!\left(\dfrac{1}{x}\right), & x\in(-1,1)\setminus\{0\} \\ \dfrac{\pi}{2}, & x=0 \end{cases}$, then $f'(0)$ is:

$\lim_{x \to 4} \frac{\cos x - \cos a}{\cos x - \cot a} = \sin^3 a$State whether this is true or false.

Evaluate: \[\lim_{x \to 3} \frac{\sqrt{3x} - 3}{\sqrt{2x - 4} - \sqrt{2}}\]

If $f(c) = 2$, $f'(c) = 1$, $g(c) = -1$, $g'(c) = 2$ then $\lim_{x \to c} \frac{g(x)f(c) - g(c)f(x)}{x - c} =$ ______

The derivative of $\tan^{-1}\left(\dfrac{\sin x - \cos x}{\sin x + \cos x}\right)$, with respect to $\dfrac{x}{2}$, where $x \in \left(0, \dfrac{\pi}{2}\right)$ is:

Limits, Continuity & Differentiability Questions (1044)