Limits, Continuity & Differentiability Questions - JEE

Let $f(x)$ be a continuous and differentiable function such that $\displaystyle\lim_{h \to 0} \frac{f(3+7h) - f(3+4h)}{h} = 4$. Then the value of $f'(3)$ equals:

$\lim_{x \to 2} \dfrac{\sqrt{2\sin^2(x-2)}}{x-2}$

$\dfrac{d}{dx}\left[\tan^{-1}\\!\left(\dfrac{\sqrt{2-x}}{1+x^2}\right)\right]$ equals $(x \ge 0)$:

Find $a$, $b$ and $c$ such that \[\lim_{x \to 0} \frac{ax e^x - b\log(1+x) + cx e^{-x}}{x^2 \sin x} = 2\]

Let $f(x) = x + \sin x$. Suppose $g$ denotes the inverse function of $f$. The value of $g'\\!\left(\frac{\pi}{4} + \frac{1}{\sqrt{2}}\right)$ has the value equal to:

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

If $y = 1 + \dfrac{c_1}{x - c_1} + \dfrac{c_2 x}{(x - c_1)(x - c_2)} + \dfrac{c_3 x^2}{(x - c_1)(x - c_2)(x - c_3)}$, then $\dfrac{dy}{dx}$ is equal to:

Let $f(x) = ax^4 + bx^3 + cx^2 + dx + e$. If $\displaystyle\lim_{x \to 0}\left(\dfrac{f(x)}{x^2} + 1\right) = 3$, $f'(1) = 0$ and $f'(2) = 0$, then the value of $a$ is:

Given that $f(a) = g(a) = k$; $f^n(a) \neq g^n(a)$ for some $n \in \mathbb{N}$ and \[\lim_{x \to a} \frac{f(a)\cdot g(x) - f(a) - g(a)f(x) + g(a)}{g(x) - f(x)} = 4\] Find the value of $k$.

Let $ f: (-1, 1) \to R $ be a differentiable function with $ f(0) = -1 $ and $ f'(0) = 1 $. Let $ g(x) = [f(2f(x)+2)]^2 $. Then $ g'(0) = $

$f(x)=[\sin x]+\sqrt{\sin x-[\sin x]}$, where [.] is GIF. $f(x)$ is:

$f:\mathbb{R}\to\mathbb{R}$, $g(x)=|f(x)|$. Which are NOT always true?

Let f and g be two functions such that g(f(x)) is defined. If f is differentiable at x and g is differentiable at f(x), then find the value of $7g'(2\pi) + 3g''(2\pi)$, given that $f'\!\left(\dfrac{3\pi}{2}\right) = \dfrac{1}{3}$ and $f''\!\left(\dfrac{3\pi}{2}\right) = 0$.

If $f(4x) = 4f(x)$ for all $x$ and $f'(1)=2$, then $\displaystyle\lim_{x\to 1}\frac{\sqrt{f(x)}-\sqrt{f(1)}}{\sqrt{x}-1}$ is equal to:

If $2y = \cot^{-1}\!\left(\sqrt{\dfrac{\sqrt{3}\cos x + \sin x}{\sqrt{3}\cos x - \sin x}}\right)$, then $\dfrac{dy}{dx}$ is equal to:

If $f(x) = \{x + \sin x\} + [x - \sin x] + [x]$ where $[y]$ and $\{y\}$ denote greatest integer function and fractional part function of $y$ respectively, then find the number of points of discontinuity in $[0, \pi]$.

If $f(x) = \log_{\sec x} |\cos 4x| + |\sin x|$, then find $\frac{dy}{dx}$ at $x = -\frac{\pi}{6}$ from the first principle.

tignum function $\text{tgn}(x)=\begin{cases}1 & [x]\text{ even}\\ -1 & [x]\text{ odd}\end{cases}$. $f(x)=\text{tgn}(x)\cdot\sin(x)\cdot|x|$. Points of discontinuity in $[0,10]$:

limt→0(11/ sin2 t + · · · + n1/ sin2 t)sin2 t

If $\lim_{x \to 0} \frac{x(1 + m\cos x) - n\sin x}{x^3} = 1$ then $m =$ ______, $n =$ ______

We have $|f(x) - f(y)| \leq (x-y)^2$ for all $x, y \in \mathbb{R}$ and $f(0) = 0$. Find $f(1)$.

The value of $ p $ and $ q $ for which the function \[ f(x) = \begin{cases} \dfrac{\sin(p+1)x + \sin x}{x}, & x 0 \end{cases} \] is continuous for all $ x $ in $ R $, is

If $f(x) = (2x-3x^2)^4 + \cos x$ and $g$ is the inverse of $f$, then which of the following is/are correct?

Let $f(a) = g(a) = k$ and their $n$th derivatives $f^n(a)$, $g^n(a)$ exist and are not equal for some $n$. Further, if $\lim_{x \to a} \dfrac{f(a)g(x) - f(a) - g(a)f(x) + g(a)}{g(x) - f(x)} = 4$, then the value of $k$ is

Given $f(x) = 5 - |x - 2|$, graph of $y = f(x)$ is as shown. So, $f(x)$ is maximum at $x = 2$, $\alpha = 2$. Given $g(x) = |x + 1|$, graph of $y = g(x)$ is as shown. So, $g(x)$ is minimum at $x = -1$, $\beta = -1$. Therefore, find $\displaystyle\lim_{x \to -\alpha\beta} \frac{(x-1)(x^2 - 5x + 6)}{x^2 - 6x + 8}$.

Evaluate $\lim_{x \to 0} \dfrac{\sqrt[3]{8+x} - \sqrt[3]{8+x^2-x^2}}{\sqrt[3]{8+x} - \sqrt[3]{8+x^2-x^2}}$Evaluate $\lim_{x \to 0} \dfrac{\sqrt{8+x} - \sqrt[3]{8+x^2} - x^2}{\sqrt[3]{8+x} - \sqrt[3]{8+x^2-x^2}}$

If $a_1 = 1$ and $a_n = n(1 + a_{n-1})$ $\forall\, n \geq 2$, and $L = \lim_{n \to \infty}\left(1 + \dfrac{1}{a_1}\right)\left(1 + \dfrac{1}{a_2}\right)\cdots\left(1 + \dfrac{1}{a_n}\right)$, then

$f(x+y)+f(x-y)=2f(x)\ \forall x,y\in\mathbb{R}$, $f(0)=0$ and $f$ is differentiable. Which are correct?

If $x = 3\tan t$ and $y = 3\sec t$, then the value of $\dfrac{d^2y}{dx^2}$ at $t = \dfrac{\pi}{4}$, is:

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

Let $g$ be the inverse of $f$. If $f(x)=x^2+3x-3$ (for appropriate domain) and $g(7)=1$, find the value of $g'(7)$. (Express as lowest fraction; if $p/q$, give $p+q$; answer 2 from key means $g'(7)=1/5$ giving $p+q=6$... or $g'(7)=2$).

Let $f$ be differentiable at $x=0$ and $f'(0)=1$. Then $\lim_{h\to 0}\dfrac{f(h)-f(-2h)}{h}=$

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

Let A = \[ A = \lim_{x \to 0} \frac{2^{\tan x} - 2^{\sin x}}{x^3} \] Find the value of $e^{4A}$.

$ \dfrac{d^2x}{dy^2} $ equals

If $\lim_{x\to 0}\dfrac{1-\cos\left(1-\cos\frac{x}{2}\right)}{2^m x^n}$ equals the left hand derivative of $e^{-|x|}$ at $x=0$, find |n+2m| divisible by?

If $x^2 + y^2 + \sin y = 4$, then the value of $\dfrac{d^2y}{dx^2}$ at the point $(-2, 0)$ is

If $t = e^{x^2}$ and $y = t^2 - 1$ then $\left(\dfrac{dy}{dx}\right)_{x=1}$ is

Let $f:\mathbb{R}\to\mathbb{R}$ be differentiable with $|f(x)-f(y)|\le|x-y|^3$ for all $x,y\in\mathbb{R}$. If $f(10)=100$, then $f(20)=$

Graph of $f(x)$ shown (piecewise linear on [0,5] with values f(0)=1, f(1)=1, f(2)=2, f(3)=2, f(4)=2, f(5)=2 approximately). Which are correct?

Given that $f(x) \geq 0$ and continuous $\forall x \in \mathbb{R}$, and $A = \int_{\pi/4}^{\beta} f(x)\, dx = \left(\beta \sin\beta + \dfrac{\pi}{4}\cos\beta + \sqrt{2}\right)\beta$, $\beta > \dfrac{\pi}{4}$. Find $f\left(\dfrac{\pi}{2}\right)$.

The value of $\lim_{x \to 0^+} \dfrac{\displaystyle\int_0^{\arctan x} \sin t^2\, dt}{x\cos x - x}$ is equal to:

If $y = \left[x + \sqrt{x^2-1}\right]^{15} + \left[x - \sqrt{x^2-1}\right]^{15}$, then $(x^2-1)\dfrac{d^2y}{dx^2} + x\dfrac{dy}{dx}$ is equal to

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 $\beta$, then $\lim_{x \to -\alpha\beta} \dfrac{(x-1)(x^2 - 5x + 6)}{x^2 - 6x + 8}$ is equal to

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 equal to:

Let $f(x)$ be a differentiable function in $[-1, \infty)$ and $f(0) = 1$ such that $\lim_{t \to x+1} \frac{t^2 f(x+1) - (x+1)^2 f(t)}{f(t) - f(x+1)} = 1$. Find the value of $\lim_{x \to 1} \frac{\ln(f(x)) - \ln 2}{x-1}$.

Which statements are correct?(A) ∃ f:[0,1]→ℝ discontinuous everywhere with |f| continuous everywhere(B) F=f·g, f diff at x=a, f(a)=0, g continuous at x=a ⟹ F diff at x=a(C) Rf'(a)=2, Lf'(a)=3 ⟹ f non-diff at x=a but always continuous(D) f(a) and f(b) have opposite signs ⟹ ∃ solution of f(x)=0 in (a,b) if f continuous on [a,b]

If the function $f$ defined on $\left(\dfrac{\pi}{6},\,\dfrac{\pi}{3}\right)$ by \[f(x) = \begin{cases} \dfrac{\sqrt{2}\cos x - 1}{\cot x - 1}, & x \neq \dfrac{\pi}{4} \\ k, & x = \dfrac{\pi}{4} \end{cases}\] is continuous, then $k$ is equal to:

Without expansion or using L'Hospital's rule, prove that $\lim_{\theta \to 0} \dfrac{3\theta - \sin 3\theta}{\theta^3} = \dfrac{1}{6}$.

If $y = (x^2+2x)(3x^4+4x^3)$, find the number of zeros of $y'$ in $(0,\infty)$.

Limits, Continuity & Differentiability Questions (1044)