Continuity Questions - JEE

Let a > 0 be a root of 2x2 + x −2 = 0. If lim x→1/a 16 1 −cos(2 + x −2x2) 1 −ax2 = α + β √ 17, where α, β ∈Z, then α + β is equal to

If $y = \log_{10} x + \log_x 10 + \log_x x + \log_{10} 10$, then $\dfrac{dy}{dx}\bigg|_{x=10} =$ (answer as integer, multiply by $10\ln 10$)

Let $f(x)=x^3+3x+1$ and $g$ be its inverse. Then $g'(5)$:

Let $f(x) = \dfrac{\sin x + \cos x - 1}{x}$ for $x\neq 0$ and $f(0)=0$. If $f$ is differentiable at $x=0$, find $f'(0)$.

If $y = \log_e\!\left(\dfrac{x^2}{e^2}\right)^2$, then $\dfrac{d^2y}{dx^2}$ at $x=e$ is:

For $x > 0$, if $(2y)^{2x} = 4e^{2x-2y}$, then $\dfrac{dy}{dx}$ is equal to:

Let $f(x) = \begin{cases} -1, & -2 \le x

Given $f(x) = \dfrac{1}{x} - \dfrac{k-1}{e^{2x}-1},\; x \neq 0$. If $f(x)$ is continuous at $x = 0$ and $f(0) = 1$, find the value of $k$.

If $f(x) = \sin^{-1}\left(\dfrac{2 \times 3^x}{1+9^x}\right)$, then $f'\left(-\dfrac{1}{2}\right)$ equals

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) $ be a non-negative continuous function such that the area bounded by the curve $ y = f(x) $, $ x $-axis and the ordinates $ x = \dfrac{\pi}{4} $ and $ x = \beta > \dfrac{\pi}{4} $ is $ \left( \beta \sin\beta + \dfrac{\pi}{4}\cos\beta + \sqrt{2}\beta \right) $. Then $ f\!\left(\dfrac{\pi}{2}\right) $ is

If the function \[ f(x) = \begin{cases} \dfrac{\sqrt{2 + \cos x} - 1}{(\pi - x)^2}, & x \neq \pi \\ k, & x = \pi \end{cases} \] is continuous at $ x = \pi $, then $ k $ equals

If $|f(x) - f(y)| \leq 2|x - y|^{3/2}$, then which of the following is true about $f'(x)$?

Let $f$ be a differentiable function in the interval $(0,\infty)$ such that $f(1)=1$ and $\displaystyle\lim_{t\to x}\frac{t^2f(x)-x^2f(t)}{t-x}=1$ for each $x>0$. Then $2f(2)+3f(3)$ is equal to

Match the number of points of discontinuity/continuity: 1) $f(x)=(\text{sgn}(x))(\text{sgn}(-x))$; 2) $f(x)=e^{|x|}-1$ if rational, $\frac{1}{1+x^2}$ if irrational; 3) $g(x)=\lim_{n\to\infty}\frac{(f(x))^n-1}{(f(x))^n+1}$ in $(0,\infty)$ where $f(x)=(x-1)(x-2)(x-3)+1$; 4) $f(x)=\max([x],|x|)$ in $\mathbb{R}$

Given $f(1) = 1$, $f'(1) = 3$. Find the derivative of $f[f(f(x))] + (f(x))^2$ at $x = 1$.

If $y = \sec(\tan^{-1} x)$, then $\dfrac{dy}{dx}$ at $x = 1$ is equal to:

$\lim_{x \to 0} \dfrac{\sqrt{1 - \cos 2x}}{\sqrt{2}\, x}$ is

Let $L_1=\displaystyle\lim_{x\to0}\frac{\tan3x-\tan x}{2\cos4x\sin3x+\sin5x-(2-2\cos4x+\sin3x)}$ and $L_2=\displaystyle\lim_{x\to0^+}\frac{e^{\sec(\tan x^{1012n})}-e}{\sin(x^{2025m})}=\frac{e}{2}$; $n,m\in\mathbb{N}$. Then $\dfrac{1}{4L_1-\frac{m}{n}}$ is

Let $ y $ be an implicit function of $ x $ defined by $ x^{2x} - 2x^x \cot y - 1 = 0 $. Then $ y'(1) $ equals

Let $\alpha$ and $\beta$ be the distinct roots of $ax^2 + bx + c = 0$, then $\lim_{x \to \alpha} \dfrac{1 - \cos(ax^2 + bx + c)}{(x - \alpha)^2}$ is equal to

Let $f(x) = \begin{cases} (x-1)^{\frac{1}{2-x}}, & x > 1,\; x \neq 2 \\ k, & x = 2 \end{cases}$. The value of $k$ for which $f$ is continuous at $x = 2$ is

For $x \to 0$, $\frac{\sin x}{x}$ is

If $y = f(x)$ is differentiable for all $x \in \mathbb{R}$, then:

If $x^2 + y^2 = r^2$ and $z = \dfrac{1}{r}$ then $z = \sqrt{1 + \left(\dfrac{dy}{dx}\right)^2}$.State whether the statement is true or false.

Let $\text{Lt}_{n \to \infty} f(n) = l$. Given that $\text{Lt}_{n \to \infty} f(n+1) = \dfrac{1}{2} \text{Lt}_{n \to \infty} \left[f(n) + \dfrac{9}{f(n)}\right]$ and $f(n) > 0 \; \forall n \in N$, find $\text{Lt}_{n \to \infty} f(n)$.

It is given that $f(x) = \text{Min}\{x+1, |x|+1\}$. Then which of the following is true?

Let BC be the diameter of a circle centred at O. Point A is variable on circumference. If BC = 1, then limA→B BM (Area of sector OAB)2 is:

$\lim_{x\to\infty} \dfrac{\left(\displaystyle\int_0^x e^{x^2}\,dx\right)^2}{\displaystyle\int_0^x e^{2x^2}\,dx} = $

The value of a for which $\lim_{x \to 0} \dfrac{\left(e^x - 1\right)^4}{\sin\!\left(\dfrac{x^2}{a^2}\right)\ln\!\left(1 + \dfrac{x^2}{2}\right)} = 8$ is

$\lim_{x \to 0} \dfrac{\tan(\pi \sin^2 x) + (|x| - \sin(x[x]))^2}{x^2}$ is equal to: (where $[\,]$ denotes greatest integer function)

Evaluate: $\lim_{n \to \infty} \dfrac{(3n+3)(3n+2)(3n+1)(3n!)}{(n+1)^3 (3n)!}$

If $f(x)$ is a continuous function $\forall x\in\mathbb{R}$ and the range of $f(x)$ is $(2,\sqrt{26})$ and $g(x)=\left[\dfrac{f(x)}{C}\right]$ is continuous $\forall x\in\mathbb{R}$, then the least positive integral value of $C$ is (where $[\cdot]$ denotes GIF)

Evaluate $\lim_{x \to \infty} \dfrac{x^a \sin\dfrac{1}{x^a} + x^2}{1 + |x|^3}$ when $x \to \infty$.

Evaluate $\lim_{x \to \pi/2} \tan x \cdot \log \sin x$.

Let $f(x) = \lim_{n \to \infty} \dfrac{x^{2n-1} + ax^3 + bx^2}{x^{2n} + 1}$ is continuous for all $x \in \mathbb{R}$. If points $A(-a, 3)$ and $B((b+1), -1)$ are points of relative maximum and minimum of a cubic polynomial $y = g(x)$, then the value of $g(2)$ is:

Continuity Questions (1086)