Limits, Continuity & Differentiability Questions - JEE

The number of values of $x$, $x \in [-2, 3]$ where $f(x) = [x^2]\sin(\pi x)$ is discontinuous is (where $[\cdot]$ denotes greatest integer function)

If f(x) = $\begin{cases} \frac{-x}{2}, & x > 0 \\ -\cos x, & 0 \leq x \leq 1 \\ \ln x, & x > 1 \end{cases}$, then find the number of points where f(x) is not differentiable.

$\lim_{x \to 0} \dfrac{x + 2\sin x}{\sqrt{x^2 + 2\sin x + 1} - \sqrt{\sin^2 x - x + 1}}$ is __________.

If $ f(x) = \begin{cases} ax + b, & x \le -1 \\ ax^3 + x + 2b, & x > -1 \end{cases} $ is differentiable for all $ x \in \mathbb{R} $, then find $ b - a = $ __________.

Let $f : R \to R$ be a continuously differentiable function such that $f(2) = 6$ and $f'(2) = \dfrac{1}{48}$. If $\displaystyle\int_0^{f(x)} 4t^3\, dt = (x - 2)\, g(x)$, then $\lim_{x \to 2} g(x)$ is equal to __________.

Find $\lim_{x \to \infty} \left( \frac{p^{1/x} + q^{1/x} + r^{1/x}}{3} \right)^h$ [where $p, q, r, a > 0]$

The value of $f(0)$ so that the function $f(x) = \frac{1 - \cos(1 - \cos x)}{x^4}$ is continuous everywhere is $k$, then value of $10k$ is

$\lim_{x \to \pi/4} \dfrac{\cot^3 x - \tan x}{\cos(x + \pi/4)}$ is __________.

200. If $y = 2\tan^{-1}\!\left(\dfrac{\sqrt{1+x^2}-1}{x}\right)$, then the value of $\dfrac{d^2y}{dx^2}$ at $x = 2$ is:

$\displaystyle\lim_{x\to\frac{\pi}{2}}\frac{\displaystyle\int_{x^3}^{(\pi/2)^3}(\sin(2t^{1/3})+\cos(t^{1/3}))\,dt}{\left(x-\dfrac{\pi}{2}\right)^2}$ is equal to

Ex. 36: Statement I: $\lim_{x \to 3/2} \frac{\sin(\cot^2 x)}{(3-2x)^2} = \frac{1}{2}$Statement II: $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$ and $\lim_{\theta \to 0} \frac{\tan \theta}{\theta} = 1$, where $\theta$ is measured in radians.

Ex. 66 The function $f'(x)$ is

Let $p = \lim_{x \to 0^+} (1 + \tan^2 x)^{1/2x}$, then $\log p$ is equal to

If $f(x) = \lim_{n \to \infty} \frac{x^{2n}-1}{x^{2n}+1}$ then $f(x)$ is discontinuous at

If f : [1, 10] → [1, 10] is a non-decreasing function and g : [1, 10] → [1, 10] is a non-increasing function. Let h(x) = f(g(x)) with h(1) = 1. Then, h(2)

If $f(x) = \begin{cases} x^2 + 1, & x

Let $[x]$ be the greatest integer $\leq x$. Then the number of points in the interval $(-2,1)$ where the function $f(x)=|[x]|+\sqrt{x-[x]}$ is discontinuous, is _____.

If $\alpha>\beta>0$ are roots of $ax^2+bx+1=0$, and $\displaystyle\lim_{x\to 1/\alpha}\left(\dfrac{1-\cos(x^2+bx+a)}{2(1-\alpha x)^2}\right)^{1/2}=\dfrac{1}{k}\left(\dfrac{1}{\beta}-\dfrac{1}{\alpha}\right)$, then $k$ is equal to

Among (S1): $\displaystyle\lim_{n\to\infty}\dfrac{1}{n^2}(2+4+6+\cdots+2n)=1$ and (S2): $\displaystyle\lim_{n\to\infty}\dfrac{1}{n^{16}}(1^{15}+2^{15}+\cdots+n^{15})=\dfrac{1}{16}$,

The value of $\dfrac{dy}{dx}$ for the curve $2y = 3 - x^2$ is

If $f(x)$ and $g(x)$ have no derivative at $x = a$, then $f(x) + g(x)$ may have a derivative at $x = a$.State whether the statement is true or false.

If $f(t) = \tan^{-1}\left[\dfrac{1}{2}(\sqrt{1+t^2}-1)\right]$ then $f'(0)$ is

274. $y = \cos^{-1}\!\left(\log_2 2^{\ln e^{\sin^{-1}\sin x}}\right)$. For $y$ as defined above, the value of $\frac{dy}{dx}$ at $x = \frac{\pi}{4}$ is:

If limα→0 ecos(αn)−e αm = −e/2, find m/n.

201. Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that for all $x, y \in \mathbb{R}$, $|f(x) - f(y)| \leq 6|x - y|^2$. If $f(3) = 6$, then $f(6)$ is equal to:

Find the largest non-negative integer a for which limx→1 n −ax+sin(x−1)+a x+sin(x−1)−1 o 1−x 1−√x = 1/4.

If $\displaystyle\lim_{x\to0}\dfrac{3+\alpha\sin x+\beta\cos x+\log_e(1-x)}{3\tan^2x}=\dfrac{1}{3}$, then $2\alpha-\beta$ is equal to:

tan(x/2 r+1 )+tan (x/2 3 r+1 ) x f (x) Let f (x) = lim . Then lim is equal to n e -e n\to\infty \sum ( ) x\to0 r=0 2 r+1 (x-f (x)) 1-tan (x/2 )

2 (2x -3x+5)(3x-1) 2 limx\to\infty 2 x is equal to : (3x +5x+4)\sqrt(3x+2)

Let [t] be the greatest integer less than or equal to t. Then the least value of p \in N for which lim x\to0 + (x ([ x1 ] + [ x2 ] + \ldots + [ xp ]) - x 2 ([ x12 ] + [ x2 2 ] + \ldots + [ x9 2 ])) \ge 1 is equal 2 2 to ________. x

Let the function, 2 -3ax - 2, x < 1 f (x) = { 2 a + bx, x \ge 1 be differentiable for all x \in R, where a > 1, b \in R. If the area of the region enclosed by y = f (x) and the line y = -20 is \alpha + \beta\sqrt3, \alpha, \beta \in Z , then the value of \alpha + \beta is ________

lim x\to0 cosec x (\sqrt2 cos 2 x + 3 cos x - \sqrtcos 2 x + sin x + 4) is:

1 If lim , then \alpha is equal to ________ t \alpha 8 3 t t\to0 (\int (3x + 5) dx) = ( ) 0 5e 5

939. If $x = 4t^3 + 3$, $y = 4 + 3t^4$ and $\dfrac{\left(\dfrac{d^2x}{dy^2}\right)}{\left(\dfrac{dx}{dy}\right)^n}$ is a constant, then find the value of $\dfrac{4}{5} + \dfrac{4}{5n} + \dfrac{4}{5n^2} + \ldots$ upto infinity.

Let $f(x)=|2x^2+5|x|-3|$, $x\in\mathbb{R}$. If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m+n$ is equal to:

Consider the function $f:(0,\infty)\to\mathbb{R}$ defined by $f(x)=e^{-|\log_e x|}$. If $m$ and $n$ be respectively the number of points at which $f$ is not continuous and $f$ is not differentiable, then $m+n$ is

Let $f(x)=\begin{cases}x-1,&x\text{ is even}\\2x,&x\text{ is odd}\end{cases}$, $x\in\mathbb{Z}$. If for some $a\in\mathbb{N}$, $f(f(f(a)))=21$, then $\displaystyle\lim_{x\to a^-}\left\{\dfrac{|x|^3}{a}-\left[\dfrac{x}{a}\right]\right\}$, where $[t]$ denotes the greatest integer less than or equal to $t$, is equal to:

Let the slope of the line $45x+5y+3=0$ be $27r_1+\dfrac{9r_2}{2}$ for some $r_1,r_2\in\mathbb{R}$. Then $\displaystyle\lim_{x\to3}\left(\int_3^x\dfrac{8t^2}{\frac{3r_2x}{2}-r_2x^2-r_1x^3-3x}\,dt\right)$ is equal to

If $\displaystyle\lim_{x\to0}\dfrac{ax^2e^x-b\log_e(1+x)+cxe^{-x}}{x^2\sin x}=1$, then $16(a^2+b^2+c^2)$ is equal to

Let $f$ be a differentiable function such that $\displaystyle\lim_{x \to 1} \frac{f(1+x^3-x)-f(x)}{\sin(x-1)} = \lim_{x \to 0} \frac{f(1-x)-f(1)}{x} + 10$, then $f'(1)$ is equal to:

We have f(x) = ex − x and g(x) = x2 − x. If f(g(x)) is an increasing function, then x belongs to

If $\lim_{t \to 0}\!\left(\int_0^1 (3x+5)^t\,dx\right)^{1/t} = \dfrac{\alpha}{5e} \cdot \left(\frac{8}{5}\right)^{2/3}$, then $\alpha$ is equal to ___

$\lim_{x \to 0} \csc x\!\left(\sqrt{2\cos^2 x + 3\cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)$ is:

Let $f(x) = \begin{cases} 3x, & x < 0 \\ \min\{1+x+[x],\, x+2[x]\}, & 0 \leq x < 2 \\ 5, & x > 2 \end{cases}$ where $[\cdot]$ denotes the greatest integer function. If $\alpha$ and $\beta$ are the number of points where $f$ is not continuous and not differentiable, respectively, then $\alpha + \beta$ equals ___

Let the function $f(x) = \begin{cases} -3ax^2 - 2, & x < 1 \\ a^2 + bx, & x \geq 1 \end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a > 1$, $b \in \mathbb{R}$. If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta\sqrt{3}$, $\alpha, \beta \in \mathbb{Z}$, then the value of $\alpha + \beta$ is ___

Let $a>0$ be a root of the equation $2x^2+x-2=0$. If $\displaystyle\lim_{x\to\frac{1}{a}}\frac{16(1-\cos(2+x-2x^2))}{(1-ax)^2}=\alpha+\beta\sqrt{17}$, where $\alpha,\beta\in\mathbb{Z}$, then $\alpha+\beta$ is equal to

If the function $f(x)=\begin{cases}\dfrac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}, & x\neq0\\ a\log_e2\log_e3, & x=0\end{cases}$ is continuous at $x=0$, then the value of $a^2$ is equal to:

Let $f:(0,\pi)\to\mathbb{R}$ be a function given by $f(x)=\begin{cases}\left(\dfrac{8}{7}\right)^{\frac{\tan8x}{\tan7x}}, & 0<x<\dfrac{\pi}{2}\\ a-8, & x=\dfrac{\pi}{2}\\ (1+|\cot x|)^{\frac{b}{a}|\tan x|}, & \dfrac{\pi}{2}<x<\pi\end{cases}$ where $a,b\in\mathbb{Z}$. If $f$ is continuous at $x=\dfrac{\pi}{2}$, then $a^2+b^2$ is equal to:

$\displaystyle\lim_{n\to\infty}\frac{(1^2-1)(n-1)+(2^2-2)(n-2)+\cdots+((n-1)^2-(n-1))\cdot1}{(1^3+2^3+\cdots+n^3)-(1^2+2^2+\cdots+n^2)}$ is equal to

997. 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.]

Limits, Continuity & Differentiability Questions (1044)