Limits, Continuity & Differentiability Questions - JEE

The value(s) of n for which limx→1 ex−1−x (x−1)n exists is/are:

Let $a = \lim_{x \to 0} \frac{\ln(\cos 2x)}{3x^2}$, $b = \lim_{x \to 0} \frac{\sin^2 2x}{x(1-e^x)}$, $c = \lim_{x \to 1} \frac{x - \sqrt{x}}{\ln x}$.Then $a$, $b$, $c$ satisfy:

The value of $\lim_{x \to \pi/4} (1 + [x])^{\frac{1}{\ln(\tan x)}}$ is:(where $[\cdot]$ denotes greatest integer function)

$f(x)=x^2\sin(1/x)$ (x≠0), f(0)=0. At x=0:

If limx→0 x(1+a cos x)−b sin x x3 = 1, then:

Match List-I with List-II and select the correct answer. List-I: (P) Let $f(x)=x^2-4x+3$. Find $g$ the inverse of $f$ and find $g'$ at $f(x)=2$. (Q) $f:R\to R$, $f(x)=x^3+x$, $x_0$ such that $f'(x_0)=3$. Find $f''(x_0)/(f'(x_0))^{3/2}$. (R) If $f(x)=x+\sin x$, $g=f^{-1}$, find $g'(\pi)$. (S) If $x^2+y^2=1$, find $yy''-(y')^2+1$. List-II: (1) $-1$ (2) 1 (3) $\tfrac{1}{2}$ (4) $-\tfrac{1}{2}$

Let $ g(x) = 6x^2 - 18x + 8 $, $ f_1(x) = |g(x)| $, $ f_2(x) = |f_1(x) - P_1| $, $ f_3(x) = |f_2(x) - P_2| $ and if $ P_1 = 7 $, then the range of $ P_2 $ such that $ f_3(x) $ has exactly 10 points of non-differentiability is:

Let \(f(x) = \begin{cases} x + 3 & -2

Let \[f(x) = \begin{cases} \dfrac{1}{\ln x} & \text{if } x > 0,\ x \neq 1 \\ \dfrac{1}{\ln(-x)} & \text{if } x The number of points of discontinuity of $f(x)$ is:

Let $f(x)=x^5+2e^{x/4}$ for all $x\in\mathbb{R}$. Consider a function $g(x)$ such that $(g\circ f)(x)=x$ for all $x\in\mathbb{R}$. Then $g'(2)$ is equal to:

Let $ [x] $ denote the greatest integer less than or equal to x. Then $ \lim_{x \to 0} \frac{\tan(\pi \sin^2 x)+\left( |x| - \sin(x[x]) \right)^2}{x^2} $

$\lim_{x \to \frac{\pi}{2}} \frac{\sin x}{\left[\frac{1}{4} - 2\cos^{-1}\left(\frac{3\sin x - \sin^3 x}{4}\right)\right]}$ (where $[\cdot]$ denotes greatest integer function) is:

The value of $\lim_{n \to \infty} \frac{[r] + [2r] + \cdots + [nr]}{n^2}$, where $r$ is a non-zero real number and $[x]$ denotes the greatest integer function, is equal to

If $\lim_{x \to 1} \frac{x^4 - 1}{x - 1} = \lim_{x \to k} \frac{x^3 - k^3}{x^2 - k^2}$, then $k$ is

The value of limx→0 −5 sin x x + 6 sin x x is (where [·] denotes G.I.F):

$\lim_{x \to 0} \frac{e^{\tan x} - e^x}{\tan x - x}$ is equal to

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

$\lim_{x \to 0} \left(\frac{e^x}{x}\right)^{\frac{x}{x}}$

If $f(x) = \begin{cases} e^{1/x} & x \neq 0 \\ 0 & x = 0 \end{cases}$ and $4x + 1 = 28$, then which statement about $f(x)$ is true?

Let $f(x) = \begin{cases} x^2 \cos\frac{1}{x} & \text{when } x \neq 0 \\ 1 & \text{when } x = 0 \end{cases}$. Then, $f(x)$ is

If the function $f(x) = \begin{cases} k_1(x - \pi)^2 - 1, & x \leq \pi \\ k_2 \cos x, & x > \pi \end{cases}$ is twice differentiable, then the ordered pair $(k_1, k_2)$ is equal to

Let y = y(x) be a function of x satisfying $y\sqrt{1-x^2} = k - x\sqrt{1-y^2}$ where k is a constant and $y\left(\frac{1}{2}\right) = -\frac{1}{4}$. Then $\frac{dy}{dx}$ at $x = \frac{1}{2}$ is equal to

Given that f(x) is continuous at x = 0, and\[\lim_{x \to 0^+} \frac{\sin^3(x) \log(1 + 3x)}{x(\tan^{-1} x)^2 (e^{5x} - 1)} = a\]Find the value of a.

Derivative of $(\log x)^{\log x}$ with respect to $x$ is

If $x^m y^n = (x+y)^{m+n}$, then $\frac{dy}{dx}$ is equal to

If $y = \tan^{-1}\!\left(\dfrac{\sqrt{1+x^2}-1}{x}\right)$ for $x>0$, then $\dfrac{dy}{dx}$ at $x=\sqrt{3}$ is:

$$\lim_{x \to a^+} \frac{x - b - \sqrt{a - b}}{\sqrt{x^2 - a^2}}$$, $$(a > b)$$ is

●Ex. 7 Let $f(x) = e^{\log x}$. If $g(x)$ is the inverse of $f(x)$, then find $g'(x)$.

$\lim_{x \to \infty} \left[(x+5)\tan^{-1}(x+5) - (x+1)\tan^{-1}(x+1)\right]$ is equal to

The value of $\lim_{n \to \infty} \sum_{i=1}^{n} a_i$ is equal to

If x2 + y2 = 1, then

Let $f(x) = \begin{pmatrix} a & -1 & 0 \\ ax & a & -1 \\ ax^2 & ax & a \end{pmatrix}$, $a\in\mathbb{R}$. Then the sum of the squares of all the values of $a$ for which $2f'(10)-f'(5)+100=0$ is:

Let $ f(x) = \dfrac{x \ln x - \ln x}{9x^2 - 2e^x x - 9x + 2e^x} + 2 $ and $ g(x) = \sin^2\!\left(\dfrac{\pi x^2}{2}\right) $, then the value of $ \lim_{x \to 1} \dfrac{f(x)}{g(x)} $ is:

Let f be a twice differentiable function such that $f''(x) = -f(x)$ and $f'(x) = g(x)$. If $h(x) = \{f(x)\}^2 + \{g(x)\}^2$, where $h(5) = 11$, find $h(10)$.

The value of $\lim_{x \to 7/2} \frac{\cot(2x - 7)}{2x^2 - 9x + 8}$ is equal to

The value of $\lim_{n \to \infty} \frac{1^3 + 2^3 + 3^3 + \ldots + n^3}{(n^2 + 1)^2}$ is

If $P(t) = \lim_{n \to \infty} \sum_{r=2}^{n} \dfrac{\sqrt{t^{2r-3}}(1-t)}{(\sqrt{t^{2r-1}}+1)(\sqrt{t^{2r-3}}+1)}$, then $P\!\left(\dfrac{1}{2}\right)$ lies in the interval:

Suppose a function $f:[0,10]\to R$ is continuous and differentiable everywhere in its domain. If $f(10) = 19$ and $|f'(x) - 5| \leq 4$ $\forall\, x$ in domain. Find maximum value of $f(0)$.

Value of $\lim_{x \to 0}\left[\dfrac{m\sin x}{x}\right]$ where $m \in I$ and [.] is GIF, is

Consider $f(x) = x \ln x$ and $g(x) = e^{2x}$. Let $a$ and $b$ be two values of $x$ satisfying $f(x) = g(x)$ with $a If \(\lim_{x \to b} \frac{f(x) - c}{g(x) - b^2} = l$, then the value of $c - l$ equals:

Let \[f(x) = \begin{cases} \max\{|x|, x^2\}, & |x| \leq 2 \\ 8 - 2|x|, & 2 S be the set of points in the interval $(-4, 4)$ at which f is not differentiable. Then S:

If $$I_1 = \lim_{x \to \infty} (\tan^{-1}(-x) - \tan^{-1}x)\cos x$$ and $$I_2 = \lim_{x \to 0} (\tan^{-1}(-x) - \tan^{-1}x)\cos x$$, then $$(I_1, I_2)$$ is

Let $f(x) = x^{\alpha - 1} \cdot \dfrac{e^{2/x} - 1}{e^{2/x} + 1}$ for $x \neq 0$ and $f(0) = 0$. For $f$ to be differentiable at $x = 0$, the value of $\alpha$ must satisfy:

Find $$\lim_{n \to \infty} \frac{1^4 + 2^4 + 3^4 + \ldots + n^4}{n^5} - \lim_{n \to \infty} \frac{1^3 + 2^3 + 3^3 + \ldots + n^3}{n^5}$$

Given \[ f(x) = \begin{cases} |x| + [x], & -1 \leq x Find the number of points where $f(x)$ is discontinuous.

Evaluate $\lim_{x \to 0^+} (\sin x)^{\tan x}$

If limx→0(1 + ax + bx2)2/x = e3, then:

Let $f(x) = \dfrac{(2^x+2^{-x})\tan x\sqrt{\tan^{-1}(x^2-x+1)}}{(7x^2+3x+1)^3}$. Then $f'(0)$ is:

Let $f(x) = \begin{cases} a & x = \frac{2}{3} \\ \frac{\sqrt{9x+3}-\sqrt{3}}{\sqrt{9-7x+4}} & x > \frac{2}{3} \end{cases}$ If $f(x)$ is continuous at $x = \frac{2}{3}$, then the value of $\frac{a}{b}$ is

[Bonus/Extra question from JEE Main 2022 set — verify in source] If $2\sin\theta = e^x + e^{-x}$, find $\dfrac{d^2y}{dx^2} + n^2y = 0$ type ODE verification. [Intentionally left for manual verification]

Limits, Continuity & Differentiability Questions (1044)