Continuity Questions (1086)

At $x = \dfrac{\pi}{4}$, $\dfrac{d}{dx}\!\left(\sin(\sin x)\right)$ equals:

Evaluate $\lim_{n \to \infty} \frac{\sin(\sqrt{n})-\sin\sqrt{n-1}}{n^1}$

$\lim_{x \to 0} \frac{x^2 + 5x}{x^2 + x^3}$

If f is a differentiable function and \[\lim_{x \to 1} \frac{f(1+x^3-x)-f(x)}{\sin(x-1)}\] exists finitely, find \(f'(1)\).

If \(f(0) = 1\) and \(\displaystyle\lim_{t \to x} \frac{\sec x \cdot f(t) - f(x) \sec t}{t - 1} = \sec^2 x\). The value of \(\dfrac{f(0)}{f'(0)}\), is:

Let \(f(x) = |x - \pi| \cdot (e^{|x|} - 1)\sin|x|\)Then which of the following is true?(The set S of points where f is not differentiable)

\(f(x)=(x^2-1)|x^2-3x+2|+\cos(|x|)\) is NOT differentiable at:

If \[ \lim_{\alpha \to 0} \frac{e^{\cos(\alpha^n)-1}}{\alpha^m} = -\frac{1}{2} \] then find the value of \(m - 2n\).

Let $$f(x)$$ be a real valued function defined for all $$x \neq -1, 1$$, satisfying $$f(1) = 1$$ and $$f'(x) = \frac{1}{x^2 + (f(x))^2}$$; then $$\lim_{x \to \infty} f(x)$$

If 0 and f(a, b) = \frac{\tan b - \tan a}{b - a}, then

Find \lim_{x \to 2} \frac{x - 2}{|x - 2|}\

If f(x) = \begin{cases} [x]-[-x] & , x \neq 2 \\ ; & , x = 2 \end{cases} and f is continuous at x=2, where [\cdot] denotes greatest integer function, then ; is

Let f(x) satisfy the requirements of Lagrange's mean value theorem in [0, 2]. If f(0) = 0 and |f'(x)| \leq \frac{1}{2} for all x \in [0, 2], then

Given f(x) = \[ \begin{cases} |x+1| & \text{if } x . Then number of point(s) of discontinuity of f(x) is-

If f(x) is continuous and f(\( \frac{9}{2} \)) = \( \frac{2}{9} \), then the value of lim \( x \to 0 \) f(\( \frac{1 - \cos 3x}{x^2} \)) is-

Let f(x) be a twice differentiable function defined on (-\infty, \infty) such that f(x) = f(2-x) and f'\left(\frac{1}{2}\right) = f'\left(\frac{1}{4}\right) = 0.The minimum number of values where f''(x) vanishes on [0, 2] is:

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

If y = (\cos x)^{(\cos x)^{(\cos x)^{\cdots \infty}}}, then \frac{dy}{dx} is equal to

Find the number of points where f(x) = [\sin x - \cos x] (where [\cdot] denotes greatest integral function), x \in [0, 2\pi] is not continuous.

\lim_{n \to 0} \left(1 - \frac{1}{2^2}\right) \left(1 - \frac{1}{3^2}\right) \cdots \left(1 - \frac{1}{n^2}\right) is equal to

161. \(\lim_{x \to \infty} x\left(\left(\dfrac{x}{x+1}\right)^x - \dfrac{1}{e}\right)\) is equal to:

If y = e^{2\sin^{-1}x}, then find \frac{(x^2-1)y'' + xy'}{y}.

Let f: \mathbb{R} \to \mathbb{R} is a function satisfying f(x + y^3) = f(x) + [f(y)]^3 for all x, y \in \mathbb{R}. If f'(0) \geq 0, then f(10) is

If f(x) = \begin{cases} \frac{\sin^3(3x) \times \log(1+3x)}{(\tan^{-1}x)^2(e^{5x}-1)x}, & x \neq 0 \\ a, & x = 0 \end{cases}\) is continuous in [0,1]\), then a equals

If $h(x) = \begin{cases} \frac{\lambda\sqrt{2x+3}}{0 \leq x \leq 3} \\ \mu x + 12, & 3 < x \leq 9 \end{cases}$ is differentiable at $x = 3$, then the value of $\lambda + \mu$ is equal to

If \lim_{n \to \infty} (\sin^{-1} x)^{n+1} = 1, find the interval in which x lies.

If $f(x) = \begin{cases} \frac{\sqrt{x+3}-2}{x-1} & 0 \leq x < 4 \\ b & x \geq 4 \end{cases}$ is continuous at $x = 4$, then value of $\frac{a}{b}$ is equal to

Let f(x) = \begin{cases} \tan\left(\frac{\pi}{4} + x\right)^{1/x} & , x \neq 0 \\ k & , x = 0 \end{cases}For what value of k is f(x) continuous at x = 0?

If $f(x) = \begin{cases} px + q & x \leq 2 \\ x^2 - 5x + 6 & 2 < x < 3 \\ a x^2 + bx + 1 & x \geq 3 \end{cases}$ is differentiable everywhere, then $|p| + |q| + \left|\frac{1}{3}\right| + |1|$ is equal to

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

g defined via running max. Non-differentiable points in (0,4):

If $\log_{10}\left(\frac{x^2}{y}\right) = 2$, then $\frac{dy}{dx} =$

\(\lim_{x \to 0} \dfrac{x \tan 2x - 2x \tan x}{(1 - \cos 2x)^2}\) equals

If $f(1) = 1,\; f'(1) = 3$, then the derivative of $f(f(f(x))) + (f(f(x)))^2$ at $x = 1$ is:

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

Let \(f(x) = \begin{cases} \sqrt{x^2 - 1}, & x \leq \sqrt{10} \\ (\sqrt{10}x - 7), & \sqrt{10}

If \(\lim_{x \to 0} \left(1 + ax + bx^2\right)^{2/x} = e^3\), then

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