Continuity Questions - JEE

$$\lim_{x \to a^+} \frac{x - b - \sqrt{a - b}}{\sqrt{x^2 - a^2}}$$, $$(a > b)$$ is

●Ex. 7 Let $f(x) = e^{\log x}$. If $g(x)$ is the inverse of $f(x)$, then find $g'(x)$.

$\lim_{x \to \infty} \left[(x+5)\tan^{-1}(x+5) - (x+1)\tan^{-1}(x+1)\right]$ is equal to

The value of $\lim_{n \to \infty} \sum_{i=1}^{n} a_i$ is equal to

If x2 + y2 = 1, then

Let $f(x) = \begin{pmatrix} a & -1 & 0 \\ ax & a & -1 \\ ax^2 & ax & a \end{pmatrix}$, $a\in\mathbb{R}$. Then the sum of the squares of all the values of $a$ for which $2f'(10)-f'(5)+100=0$ is:

Let $ f(x) = \dfrac{x \ln x - \ln x}{9x^2 - 2e^x x - 9x + 2e^x} + 2 $ and $ g(x) = \sin^2\!\left(\dfrac{\pi x^2}{2}\right) $, then the value of $ \lim_{x \to 1} \dfrac{f(x)}{g(x)} $ is:

Let f be a twice differentiable function such that $f''(x) = -f(x)$ and $f'(x) = g(x)$. If $h(x) = \{f(x)\}^2 + \{g(x)\}^2$, where $h(5) = 11$, find $h(10)$.

The value of $\lim_{x \to 7/2} \frac{\cot(2x - 7)}{2x^2 - 9x + 8}$ is equal to

The value of $\lim_{n \to \infty} \frac{1^3 + 2^3 + 3^3 + \ldots + n^3}{(n^2 + 1)^2}$ is

If $P(t) = \lim_{n \to \infty} \sum_{r=2}^{n} \dfrac{\sqrt{t^{2r-3}}(1-t)}{(\sqrt{t^{2r-1}}+1)(\sqrt{t^{2r-3}}+1)}$, then $P\!\left(\dfrac{1}{2}\right)$ lies in the interval:

Suppose a function $f:[0,10]\to R$ is continuous and differentiable everywhere in its domain. If $f(10) = 19$ and $|f'(x) - 5| \leq 4$ $\forall\, x$ in domain. Find maximum value of $f(0)$.

Value of $\lim_{x \to 0}\left[\dfrac{m\sin x}{x}\right]$ where $m \in I$ and [.] is GIF, is

Consider $f(x) = x \ln x$ and $g(x) = e^{2x}$. Let $a$ and $b$ be two values of $x$ satisfying $f(x) = g(x)$ with $a If \(\lim_{x \to b} \frac{f(x) - c}{g(x) - b^2} = l$, then the value of $c - l$ equals:

Let \[f(x) = \begin{cases} \max\{|x|, x^2\}, & |x| \leq 2 \\ 8 - 2|x|, & 2 S be the set of points in the interval $(-4, 4)$ at which f is not differentiable. Then S:

If $$I_1 = \lim_{x \to \infty} (\tan^{-1}(-x) - \tan^{-1}x)\cos x$$ and $$I_2 = \lim_{x \to 0} (\tan^{-1}(-x) - \tan^{-1}x)\cos x$$, then $$(I_1, I_2)$$ is

Let $f(x) = x^{\alpha - 1} \cdot \dfrac{e^{2/x} - 1}{e^{2/x} + 1}$ for $x \neq 0$ and $f(0) = 0$. For $f$ to be differentiable at $x = 0$, the value of $\alpha$ must satisfy:

Find $$\lim_{n \to \infty} \frac{1^4 + 2^4 + 3^4 + \ldots + n^4}{n^5} - \lim_{n \to \infty} \frac{1^3 + 2^3 + 3^3 + \ldots + n^3}{n^5}$$

Given \[ f(x) = \begin{cases} |x| + [x], & -1 \leq x Find the number of points where $f(x)$ is discontinuous.

Evaluate $\lim_{x \to 0^+} (\sin x)^{\tan x}$

If limx→0(1 + ax + bx2)2/x = e3, then:

Let $f(x) = \dfrac{(2^x+2^{-x})\tan x\sqrt{\tan^{-1}(x^2-x+1)}}{(7x^2+3x+1)^3}$. Then $f'(0)$ is:

Let $f(x) = \begin{cases} a & x = \frac{2}{3} \\ \frac{\sqrt{9x+3}-\sqrt{3}}{\sqrt{9-7x+4}} & x > \frac{2}{3} \end{cases}$ If $f(x)$ is continuous at $x = \frac{2}{3}$, then the value of $\frac{a}{b}$ is

[Bonus/Extra question from JEE Main 2022 set — verify in source] If $2\sin\theta = e^x + e^{-x}$, find $\dfrac{d^2y}{dx^2} + n^2y = 0$ type ODE verification. [Intentionally left for manual verification]

Which of the following function(s) is/are discontinuous at x = 0 ?

lim n→∞ (12 −1)(n −1) + (22 −2)(n −2) + · · · + ((n −1)2 −(n −1)) · 1 (13 + 23 + · · · + n3) −(12 + 22 + · · · + n2) is equal to

If f: \mathbb{R} \to \mathbb{R}\) is a function defined by f(x) = [x]\cos\left(\frac{2x-1}{2}\pi\right)\), where [x]\) denotes the greatest integer function, then f is

The value of f at x = 0, so that function f(x) = \frac{2^x - 2^{-x}}{x}\), x \neq 0\) is continuous at x = 0, is

If $$f(x) = 0$$ is a quadratic equation such that $$f(-\pi) = f(\pi) = 0$$ and $$f\left(\frac{3\pi}{2}\right) = -\frac{3\pi}{4}$$, then $$\lim_{x \to -\pi} \frac{f(x)}{\sin(\sin x)}$$ is equal to

If f : ℝ → ℝ is the function defined by f(x) = (ex − e−x)/(ex + e−x), then

Let $f:[−1,3]\to\mathbb{R}$ be defined as $f(x)=\begin{cases}|x|+[x], & -1\leq x where \([t]$ denotes the greatest integer less than or equal to $t$. Then $f$ is discontinuous at

Let $f(x) = \int_0^x t\ln(1+t^2)\, dt$, then $f''(0)$ is:

If $2^x + 2^y = 2^{x+y}$, then $\dfrac{dy}{dx}$ is equal to which of the following expressions?

Let f : R → R be a continuous function satisfying $ f(x) + \int_0^x t f(t)\, dt + x^2 = 0 $ ∀ x. Then:

Let $h(x) = \min\{x, x^2\}$ for every real number $x$. Then,

Let \[S = \{(\lambda, \mu) \in R \times R : f(t) = (|\lambda|e^{|t|} - \mu)\sin(2|t|),\, t \in R,\] is a differentiable function\}. Then $S$ is a subset of

If $f(x) = \begin{cases} 2x^2 + 3 & x > 3 \\ ax^2 + bx + 1 & x \leq 3 \end{cases}$ is differentiable everywhere, then $a$ is equal to

252. The number of points where $f(x) = |x + [x]| - 3[2x] + 4[3x]$ is discontinuous in $[-1, 1]$, is: [Note: $[k]$ denotes greatest integer less than or equal to $k$.]

The function $f(x) = \max\{(1-x), (1+x), 2\}$ $\forall x \in \mathbb{R}$ is

If $y = \tan^{-1}\left(\frac{1}{1+x^2}\right) + \tan^{-1}\left(\frac{2x}{1-x^2}\right)$ ($y > 0$), then $\frac{dy}{dx}$ is equal to

If $a = \sec f$ and $b = \csc f$, where $f$ is a parameter, then the value of $\frac{dy}{dx}$ when $x = -\frac{3}{4}$ is

If $y = (1+x^2)^n + \sin^{-1}(\sin^2 x)$, then $\frac{d^2y}{dx^2}$ at $x = 0$ is

Let $f(x) = x\left[\frac{x}{2}\right]$, for \(-10

Let h(x) = 2 − |x − 1| and g(x) = h(|x|) + |h(x)|. Find the number of points where g(x) is non-differentiable.

The value of $\lim_{x \to \infty} \frac{ax^2 + bx + c}{dx + e}$ ($a, b, c, d, e \in \mathbb{R} - \{0\}$) depends on the sign of:

If $f(x) = 2x^3 - 3x^2 + a$ is a one-to-one function, the value of $a$ is arbitrary. Given $f(g(x))=x$, then $g'\!\left(\dfrac{11}{4}\right)$ is:

\[ \lim_{x \to 0} \frac{\sin{(\pi \cos{x})}}{x^2} \]

Find $\frac{dy}{dx}$ at $x = 3$ where $y = \sqrt{x^2 + 16}$ and $s = \frac{x}{y}$

If $x = \frac{1-\tan^2\theta}{1+\tan^2\theta}$ and $y = \frac{2\tan\theta}{1+\tan^2\theta}$, find $\frac{dy}{dx}$

25. If $y = (x + \sqrt{1 + x^2})^n$, then the value of $(1 + x^2)\dfrac{d^2y}{dx^2} + x\dfrac{dy}{dx}$ is equal to:

Continuity Questions (1086)