Continuity Questions - JEE

If $2y = \left(\cot^{-1}\left(\dfrac{\sqrt{3}\cos x + \sin x}{\cos x - \sqrt{3}\sin x}\right)\right)^2$, $x \in \left(0, \dfrac{\pi}{2}\right)$, then $\dfrac{dy}{dx}$ is equal to:

Let $f(\theta) = \frac{3}{\tan^2 \theta} \left\{(1 + \tan \theta)^3 + (2 + \tan \theta)^3 + \ldots + (10 + \tan \theta)^3\right\} - 10\tan\theta$.Then, $\lim_{\theta \to 0} f(\theta)$ is equal to

The values of a and b so that the function \[f(x) = \begin{cases} a^2 \sin x & 0 \leq x \leq \pi/4 \\ 2x \cot x + b & \pi/4 \leq x \leq \pi/2 \\ a \cos 2x - b \sin x & \pi/2 \leq x \leq \pi \end{cases}\]is continuous for $x \in [0, \pi]$, are

Let f(x) = ||x| − 1|, then points where f(x) is not differentiable is/are

$\lim_{x \to \infty} \left(\frac{x+6}{x+1}\right)^{x+4} =$ ______

$\lim_{h \to 0} \frac{f(2h + 2 + h^2) - f(2)}{f(h + h^2 + 1) - f(1)}$, given that $f'(2) = 6$ and $f'(1) = 4$ [2003 AIEEE]

The range of values of 'a' such that $\sqrt{x} = x^2 + a$ is satisfied for maximum number of values of 'x'

If f(x) is continuous such that |f(x)| ≤ 1, ∀x ∈ ℝ and $\frac{e^{f(x)} - e^{|f(x)|}}{e^{f(x)} + e^{|f(x)|}} ≥ 0$, then range of g(x) is $g(x) = \frac{e^{f(x)} - e^{-|f(x)|}}{e^{f(x)} + e^{-|f(x)|}}$

If (a + 2b cos x)(a − 2b cos y) = a2 − b2, where a > b > 0, then dy/dx at (π/4, π/4) is (JEE Main 2020)

Let $f(x+y) = f(x) + f(y) + 2xy - 1$ for all $x, y \in \mathbb{R}$. If $f(x)$ is differentiable and $f'(0) = \sin f$, then

Let f(x) be the inverse of the function g(x) and f'(x) = 1/(5·g(x)^5), then d/dx[f(x)] is equal to

Ex. 9 A function $f: \mathbb{R} \to \mathbb{R}$ satisfies the equation $f(x+y) = f(x) + f(y)$ for all $x, y \in \mathbb{R}$, $f(x) \geq 0$. Suppose that the function is differentiable at $x = 0$ and $f'(0) = 2$. Then,

Let $[\cdot]$ denote the greatest integer function and $f(x)=\displaystyle\lim_{n\to\infty}\frac{1}{n^3}\sum_{k=1}^n\left[\frac{k^2}{3^x}\right]$. Then $12\displaystyle\sum_{j=1}^\infty f(j)$ is equal to _____.

Let f(x) = e^x, g(x) = \sin^{-1} x and h(x) = f[g(x)], then \frac{h'(x)}{h(x)} is equal to

Let $[\cdot]$ represent the greatest integer function less than or equal to $x$. The value of $\lim_{x \to 0} \frac{\ln \sin x}{[\ln \tan x]} - \frac{\ln \tan x}{[\ln \tan x]}$ is

Find the value of $\lim_{x \to \frac{\pi}{2}} \frac{\cos x}{\left(1 - \tan \frac{x}{2}\right)(1 - \sin x)}\left(1 + \tan \frac{x}{2}\right)(\pi - 2x)$

If $f(x) = x^3 \text{sgn}(x)$, then

The value of $\lim_{x \to 0} \frac{\sin[x]}{[x]}$ (where $[\cdot]$ denotes the greatest integer function) is

$\frac{d}{dx}\left[\sin^{-1}\left(x\sqrt{1-x} - \sqrt{x}\sqrt{1-x^2}\right)\right]$ is equal to

If f(x) = |1 − x|, then the points where sin−1(f(|x|)) is non-differentiable, are

●Ex. 6 Let $f(x) = x + \sin x$. Suppose $g$ denotes the inverse function of $f$. Then, find the value of $g'\left(\frac{\pi}{2}\right)$.

Ex. 37: Statement I: $\lim_{m,n \to \infty} \sin(2\pi n! \cdot 3^n) = 0$ when $x$ is rational.Statement II: When $n \to \infty$ and $x$ is rational $x = \frac{p}{q}$ where $p, q$ are integers and $q \neq 0$, then $n! \cdot x = n! \cdot \frac{p}{q}$ is an integer.

$\lim_{x \to 2} \frac{3^x + 3^{3-x} - 12}{3^{-x/2} - 3^{1-x}}$ is equal to ……… .

If $\displaystyle\lim_{x \to 0}\left(\dfrac{\sin 3x}{x^3} + \dfrac{a}{x^2} + b\right) = 0$, then the value of $(a + b)$ equals:

Let $f(x) = 1 + 4x - x^2,\ \forall x \in R$ \[g(x) = \begin{cases} \max\{f(t);\ x \le t \le (x+1),\ 0 \le x

The function $f(x) = \frac{\tan(\pi[x-\pi])}{1+[x]^2}$ where $[x]$ is the greatest integer function,

Given: $f(x) = \sin^{-1}\left(\dfrac{2 \cdot 3^x}{1+(3^x)^2}\right)$. Find $f'\!\left(-\dfrac{1}{2}\right)$.

Let f(x) be defined as follows:\[f(x) = \begin{cases} b - x & -2 \le x If f is differentiable at x = −1, find the value of b − a.

If $\lim_{x \to \infty} 4x\left(\frac{\pi}{4} - \tan^{-1}\frac{x+1}{x+2}\right) = y^2 + 4y + 5$, then the product of all possible value of $y$ is ______.

Let $f(x) = \dfrac{x}{1+|x|}$. Then $f(x)$ is

53. If $f'(x) + (f'(x))^2 + (f'(x))^3 + (f'(x))^4 + \cdots \infty = e^x$, where $f'(x) \in (-1, 1)$ and $f(0) = 0$, then the value of $\displaystyle\lim_{x \to 0} (1 + f(x))^{\frac{1}{x}}$ is equal to:

If $y = e^x\sin x$, then $\dfrac{d^{16}y}{dx^{16}}$ at $x=0$ divided by $e^0$ equals:

Let f be continuous on R. If f(0) = 1 and f is defined as f(1/4n) involves sin en and exponential terms, and f is continuous on R, find f(0).

$\displaystyle\lim_{x\to0}\dfrac{e^{4|\sin x|}-2|\sin x|-1}{x^2}$

If $\lim_{n \to \infty} \left( \frac{(n^3+1)(n^3+2^3)(n^3+3^3)\ldots(n^3+n^3)}{n^{3n}} \right)^{1/n} = 4e^{3/4}e^{-b}$ (where $a, b \in \mathbb{N}$) then $a + b$ is ______.

If $\lim_{x \to \infty} \left(1 + \dfrac{a}{x} + \dfrac{b}{x^2}\right)^{2x} = e^2$, then the values of $a$ and $b$, are

Evaluate $\lim_{n \to \infty} \dfrac{1 \cdot \Sigma r + 2 \cdot \Sigma r + 3 \cdot \Sigma r + \ldots + n \cdot 1}{n^4}$

The value of $\lim_{n \to \infty} \left( \frac{1}{\sqrt{n^2}} + \frac{1}{\sqrt{n^2+1}} + \ldots + \frac{1}{\sqrt{n^2+2n}} \right)$ is ______.

Let $y$ be an implicit function of $x$ defined by $x^{2x} - 2x^x \cot y - 1 = 0$. The value of $y'(1)$, where $y'$ denotes the first derivative of $y$, is:

If $y = \sqrt{1 - \log_e\!\sqrt{x^2+1}}$, find the value of $\left.\dfrac{dy}{dx}\right|_{x=0}$.

$y = \cos^{-1}\!\left(\log_2 2^{\ln e^{\sin^{-1}\sin x}}\right)$. For $y$ as defined above, the value of $\dfrac{dy}{dx}$ at $x = \dfrac{\pi}{4}$ is:

The function $f(x) = 1$, if $x$ is rational$= 0$, if $x$ is irrationalis discontinuous at all points $x$.State whether this statement is true or false.

Let $f: R \to R$ be a differentiable function satisfying $f'(3) + f'(2) = 0$. Then $\lim_{x \to 0} \left(\dfrac{1 + f(3+x) - f(3)}{1 + f(2-x) - f(2)}\right)^{\frac{1}{x}}$ is equal to:

For continuity at $x = 0$, if $\lim_{x \to 0} \left[\dfrac{(e^x - 1)^2}{\sin\left(\dfrac{x}{k}\right) \cdot \ln\left(1 + \dfrac{x}{4}\right)}\right] = 12$, find $k$.

A function $f(x)$ satisfies the relation $f(x+y) = f(x) + f(y) + xy(x+y)$, $\forall x, y \in \mathbb{R}$. If $f'(0) = -1$, then $f'(3) = $ ______.

The inverse function of a continuous function is continuous.State whether this statement is true or false.

For $x \in \mathbb{R}$, $\lim_{x \to \infty} \left(\dfrac{x-3}{x+2}\right)^x$ is equal to

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:

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

If $g(x) = (x^2 + 2x + 3)f(x)$, $f(0) = 5$ and $\displaystyle\lim_{x \to 0}\left(\dfrac{f(x)-5}{x}\right) = 4$, then $g'(0)$ is equal to:

Continuity Questions (1086)