Differentiability Questions - JEE

limx→0 3x2+2 7x2+2 1/x2 is equal to:

237. Let $m$ be a positive integer. If $\displaystyle\lim_{x \to 0} |\cos x + \sin 2x + \sin 3x|^{\cot x} = e^m$, then the value of $m$ is:

The number of points in (1, 3), where $f(x) = a[x^2]$, $a > 1$, is not differentiable, where [x] denotes the integral part of x.

The value of limx→1 3√ 7+x3− √ 3+x2 x−1 is:

Evaluate: \[\lim_{x \to \frac{\pi}{2}} \frac{\cot x(1 - \sin x)}{-8\left(x - \dfrac{\pi}{2}\right)^3}\]

$\lim_{x \to \pi/2} \dfrac{\left[1 - \tan\left(\dfrac{x}{2}\right)\right][1 - \sin x]}{\left[1 + \tan\left(\dfrac{x}{2}\right)\right][\pi - 2x]^3}$ is

limx→0 tan π 4 + x 1/x is equal to:

limx→∞ 11/x+21/x+31/x+···+n1/x n nx where n ∈N is equal to:

Let f : R → R be a continuous function satisfying $ f(x) + \int_0^x t f(t)\, dt + x^2 = 0 $ ∀ x. Then:

Differentiable on ℝ. Find $48(a+b)$.

If $f(x)=\lim_{n\to\infty}\left(\prod_{i=1}^n\cos\frac{x}{2^i}\right)$ then $f'(x)$ is equal to

lim x→1 (5x + 1)1/3 −(x + 5)1/3 (2x + 3)1/2 −(x + 4)1/2 = m √ 5 n(2n)2/3 , where gcd(m, n) = 1. Then 8m + 12n is equal to

limx→∞ (√3x+1+√3x−1)6+(√3x+1−√3x−1)6 (x+ √ x2−1)6+(x− √ x2−1)6 · x3

If limx→0 sin x+aex+be−x+c ln(1+x) x3 is finite, then a −b + c is:

Find f(x) = limn→∞(cos x √n)n.

Let $[\cdot]$ denote the greatest integer function, and let $f(x)=\min\{\sqrt{2}x,\,x^2\}$. Let $S=\{x\in(-2,2): g(x)=|x|\,[x^2]$ is discontinuous at $x\}$. Then $\displaystyle\sum_{x\in S}f(x)$ equals

If $f(x) = \begin{cases} \dfrac{a|x|+x^2-2\sin|x|\cos|x|}{x} & x\neq0 \\ b & x=0 \end{cases}$ is continuous at $x=0$, then $a+b$ is equal to

If $f(x) = \begin{cases} \frac{[(a-n)nx - \tan x]\sin nx}{x^2} & \text{at } x = 0 \\ 0 & \text{at } x = 0 \end{cases}$, where $n$ is a non-zero real number, and $f$ is continuous at $x = 0$, then $a$ is equal to:

Let $f$ be a function defined on $(-\pi/2, \pi/2)$ as follows: $f(x) = \begin{cases} \frac{2^{[1/n]} - [x] - \frac{[x]}{[n2-1]}}{x\tan x} & x \neq 0 \\ k & x = 0 \end{cases}$. The value of $k$ so that $f$ is continuous at $x = 0$ is:

If $f(x)=x(\sqrt{x}-\sqrt{x+1})$, then:

If $f(x)=\text{sgn}(x^5)$, which are false?

Let $S$ be the set of all points where $\sqrt[5]{x^2|x|^3}-\sqrt[3]{x^2|x|}-1$ is not differentiable. $S$ is a subset of:

Number of points where $f(x)=|x-\text{sgn}(x)|$ is non-differentiable (sgn(.) denotes signum function):

Let $[t]$ denote the greatest integer less than or equal to $t$. If the function $f(x) = \begin{cases} b^2\sin\!\left(\dfrac{\pi}{2}\!\left[\dfrac{\pi}{2}(\cos x+\sin x)\cos x\right]\right) & x < 0 \\[6pt] \dfrac{\sin x - \tfrac{1}{2}\sin 2x}{x^3} & x > 0 \\[4pt] a & x = 0 \end{cases}$ is continuous at $x=0$, then $a^2+b^2$ is equal to

Let $f(x)=\begin{cases}x^3+2x^2 & x\in\mathbb{Q}\\ -x^3+2x^2+ax & x\notin\mathbb{Q}\end{cases}$. Integral value of $a$ so that $f$ is differentiable at $x=1$:

If $f(x) = |\cos x - \sin x|$, then $f'\!\left(\dfrac{\pi}{6}\right)$ equals (give answer as integer after multiplying by $-1$ if negative):

If $y = \sqrt{\dfrac{1-x}{1+x}}$, then $\dfrac{dy}{dx}$ equals:

$f(x)=\begin{cases}\frac{\ln\cos x}{ax} & x>0\\ 0 & x=0\\ \frac{e^{x^2}-1}{bx} & x. If \(f'(0)=\frac{1}{4}$, then:

Which function is continuous everywhere in its domain but has at least one point where it is not differentiable?

Let $f(x)=\begin{cases}2-|x^2+5x+6| & x\ne-2\\ b^2+1 & x=-2\end{cases}$ has relative maximum at $x=-2$. Then complete set of values $b$ can take is

Given function $f(x) = x^3 + x^2 f'(1) + xf''(2) + f'''(3)$. Then $f(2)$ equals:

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

\lim_{x \to \infty} \frac{\left(\sqrt{3x+1}+\sqrt{3x-1}\right)^6 + \left(\sqrt{3x+1}-\sqrt{3x-1}\right)^6}{\left(x+\sqrt{x^2-1}\right)^6 + \left(x-\sqrt{x^2-1}\right)^6} \cdot x^3

Given equation is $\ln(x+y) = 2xy$. Find $\frac{dy}{dx}$ at point $(0,1)$.

Given, $y = z^x$, then $y = z^y$. Find $\frac{dy}{dx}$.

\[\lim_{x \to 0} \frac{\tan(\pi \sin^2 x) + (|x| - \sin(x[x]))^2}{x^2}\] is equal to: (where $[\,]$ denotes greatest integer function)

$\lim_{x \to 0} \frac{f(x) \cdot g(x)}{x(1-g(x))}$ will be

If $f(x) = \cot^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right)$ and $g(x) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$, then $$\lim_{x \to a} \frac{f(x) - f(a)}{g(x) - g(a)}, \quad 0

If $x + |y| = 2y$ then $y$ as a function of $x$ is

197. Let $g(x) = \dfrac{1}{f^{-1}(x)}$. Given the following data:$x$01234$f(x)$$-2$$-1$246$f'(x)$1/22/314/35/3The value of $g'(4)$ is:

If $f(x) = \begin{vmatrix} x & x^2 & x^3 \\ 1 & 2x & 3x^2 \\ 0 & 2 & 6x\end{vmatrix}$, find $f'(1)$.

Let $f: R \to R$ be a function defined as \[f(x) = \begin{cases} 5, & \text{if } x \leq 1 \\ a + bx, & \text{if } 1

If the function \[f(x) = \begin{cases} x^3, & -2 \leq x where $f(x) = \sin(x-2) + a\cos(x-2)$is continuous and differentiable in $(4, 6)$, then find the range of $a$.

Let f(x) = P10 r=1 r sin rx x . The value of limx→0 f(x) is:

Let $f(x) = \dfrac{\tan x}{x}$, then the value of $\lim_{x \to 0}\left([f(x)] + x^2\right)^{\frac{1}{\{f(x)\}}}$ is equal to(where [.] and {.} denote greatest integer function and fractional part functions respectively)

Given limit = $\displaystyle\lim_{x \to \frac{\pi}{4}} \dfrac{f(\sec^2 x) \cdot 2\sec^2 x \tan x}{2x} = \dfrac{8}{\pi} f(2)$. If $k^a = 8^2$, find the value of $k^a$.

$f(x)=4|2x+3|+9[x+\frac{1}{2}]-12[x+20]$. Non-diff in (-20,20):

Let $\{x\}$ denote the fractional part of $x$ and $f(x)=\dfrac{\cos^{-1}(1-\{x\}^2)\sin^{-1}(1-\{x\})}{\{x\}-\{x\}^3}$, $x\neq0$. If $L$ and $R$ respectively denote the left hand limit and the right hand limit of $f(x)$ at $x=0$, then $\dfrac{32}{\pi^2}(L^2+R^2)$ is equal to

The value of limn→∞(4n + 5n)1/n is:

55. If \(f(x) = \begin{cases} -e^{-x} + k, & x \leq 0 \\ e^x + 1, & 0

Differentiability Questions (1063)