Continuity Questions - JEE

Consider the function $f(x) = \tan^{-1}\left(\frac{x^2-1}{x^2+1}\right), \forall x \geq 0$. If $g(x)$ is the inverse function of $f(x)$, then the value of $g'\left(\frac{\pi}{4}\right)$ is equal to

The integer $n$ for which $\lim_{x \to 0} (\sin x)^{1/x}$ is a finite non-zero number, is [2002 AIEEE]

If $I_n = \frac{d^n}{dx^n}(x^n\ln x)$, then the value of $\frac{d}{dx}(I_7 - 7I_6)$ is equal to

198. If $y = \sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\cdots}}}}}}}$, then the value of $\dfrac{dy}{dx}$ at $x = 8$ is:

The total number of points of non-differentiability of $f(x) = \max\left\{\sin^2 x, \cos^2 x, \frac{3}{4}\right\}$ in $[0, 10\pi]$, is

If $\lim_{x \to \frac{\pi}{2}} \frac{\sin x}{x}$ exists finitely, then the value of $a$ is

The value of $\lim_{x \to \pi} \left[1 - \cos\left(\frac{x}{2}\right) - \cos\left(\frac{x}{4}\right) + \cos\left(\frac{x}{2}\right) \cdot \cos\left(\frac{x}{4}\right)\right]$ is $\frac{\lambda}{4}$, then the value of $900\lambda$ is equal to there, $\lambda > 0$)

If $\lim_{x \to 0} \frac{\sin 3x}{\sin 4x} - 3$, then the value of $272 \cdot \frac{ab}{cd}$ is equal to

The value of $\lim_{x \to 0} \frac{\sin\left(\frac{x}{3}\right) \sin\left(\frac{x}{3}\right)}{\left(\frac{x}{3}\right)}$ is equal to

If $x = t^3 + t^5$ and $y = \sin t$, then $\dfrac{d^2y}{dx^2}$ equals:

Let $f(x) = x^2 - 4x - 3, x > 2$ and $g(x)$ is the inverse of $f(x)$. Then the value of $\frac{1}{g'(2)}$, where $f(z) = 2$ is (here, $g'$ represents the first derivative of $g$)

Find $\displaystyle\lim_{x \to \infty} \frac{\displaystyle\int_0^x e^{t^2}\, dt}{\displaystyle\int_0^x e^{2t^2}\, dt}$.

198. If $y = \sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\cdots}}}}}}}$, then the value of $\dfrac{dy}{dx}$ at $x = 8$ is:

Find $\lim_{x \to 4} \frac{(1-\sin(\pi x))}{(x-4)^2}$

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 $f$ be continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a)/a = f(b)/b$. Then by Rolle's theorem applied to $g(x) = f(x)/x$, there exists $x_0 \in (a, b)$ such that:

196. Let f and g be defined such that $f'(x) = f^2(x) + g^2(x)$ and $g'(x) = 2f(x)g(x) + 1$. If $f(0) = \dfrac{1}{5}$, $g(0) = \dfrac{4}{5}$, then the value of $f\!\left(\dfrac{\pi}{12}\right) + g\!\left(\dfrac{\pi}{12}\right)$ equals:

Consider the following statements about positive functions $f(x)$ and $g(x)$ whose limits to infinity exist.Statement-A: $\lim_{x\to\infty} f(x) = \lim_{x\to\infty} g(x)$Statement-B: $\lim_{x\to\infty}(f(x)-g(x))=0$Statement-C: $\lim_{x\to\infty}\sqrt{f(x)} = \lim_{x\to\infty}\sqrt{g(x)}$How many of the following six statements are true: $A\Rightarrow B,\; B\Rightarrow C,\; C\Rightarrow A,\; A\Rightarrow C,\; B\Rightarrow A,\; C\Rightarrow B$?

The value of $$\lim_{x \to 0^+} \frac{\int_1^{\cos x} (\cos^{-1} t) \, dt}{2x - \sin 2x}$$ is equal to:

If $x = e^t \cos t$, $y = e^t \sin t$ then $\dfrac{d^2 y}{dx^2}$ = ______

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

Let $f(x) = x \cos x + 2$ and $g(x)$ be the inverse function of $f(x)$, then $g'(3)$ equals to ……… .

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:

$\lim_{x \to 0} \dfrac{x \cot(4x)}{\sin^2 x \cot^2(2x)}$ is equal to __________.

797. 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.

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

Let $f:\mathbb{R}\to\mathbb{R}$ be defined as $f(x)=\displaystyle\lim_{p\to\infty}\frac{\sin\!\left((2x-1)\frac{\pi}{2}\right)\!x^{4p}e^{x^2-1}+x^{4p}}{1+x^{4p+2}-x^{4p}}$. $f(x)$ is continuous for all $x$ in

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.

Continuity Questions (1086)