Continuity Questions (1086)

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

If \(\lim_{x \to 1} \frac{x^4 - 1}{x - 1} = \lim_{x \to k} \frac{x^3 - k^3}{x^2 - k^2}\), then \(k\) is

The value of limx→0  −5 sin x x  + 6 sin x x 
is (where [·] denotes G.I.F):

$\lim_{x \to 0} \frac{e^{\tan x} - e^x}{\tan x - x}$ is equal to

$\lim_{x \to 0} \frac{e^{\sin x} - (1 + \sin x)}{x^2}$

$\lim_{x \to 0} \left(\frac{e^x}{x}\right)^{\frac{x}{x}}$

If \(f(x) = \begin{cases} e^{1/x} & x \neq 0 \\ 0 & x = 0 \end{cases}\) and \(4x + 1 = 28\), then which statement about \(f(x)\) is true?

Let \(f(x) = \begin{cases} x^2 \cos\frac{1}{x} & \text{when } x \neq 0 \\ 1 & \text{when } x = 0 \end{cases}\). Then, \(f(x)\) is

If the function \(f(x) = \begin{cases} k_1(x - \pi)^2 - 1, & x \leq \pi \\ k_2 \cos x, & x > \pi \end{cases}\) is twice differentiable, then the ordered pair \((k_1, k_2)\) is equal to

Let y = y(x) be a function of x satisfying \(y\sqrt{1-x^2} = k - x\sqrt{1-y^2}\) where k is a constant and \(y\left(\frac{1}{2}\right) = -\frac{1}{4}\). Then \(\frac{dy}{dx}\) at \(x = \frac{1}{2}\) is equal to

If f(x) = \( \begin{cases} -4\sin x + \cos x & \text{for } x \leq -\frac{\pi}{2} \\ a \sin x + b & \text{for } -\frac{\pi}{2} < x < \frac{\pi}{2} \\ \cos x + 2 & \text{for } x \geq \frac{\pi}{2} \end{cases} \) is continuous then :

Given that f(x) is continuous at x = 0, and\[\lim_{x \to 0^+} \frac{\sin^3(x) \log(1 + 3x)}{x(\tan^{-1} x)^2 (e^{5x} - 1)} = a\]Find the value of a.

Derivative of \((\log x)^{\log x}\) with respect to \(x\) is

If \(x^m y^n = (x+y)^{m+n}\), then \(\frac{dy}{dx}\) is equal to

If $y = \tan^{-1}\!\left(\dfrac{\sqrt{1+x^2}-1}{x}\right)$ for $x>0$, then $\dfrac{dy}{dx}$ at $x=\sqrt{3}$ is: