Continuity Questions - JEE

If $a=\displaystyle\lim_{x\to0}\dfrac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2}}{x^4}$ and $b=\displaystyle\lim_{x\to0}\dfrac{\sin^2x}{\sqrt{2}-\sqrt{1+\cos x}}$, then the value of $ab^3$ is:

If $\lim_{x \to \infty} \frac{px + q}{qx + p} = 1$ and $\lim_{x \to \infty} \frac{px + q}{qx + p} = m$, where $p, q \neq 0$, then $\lim_{x \to 0}$ is

If $e^y + xy = e$, the ordered pair $\!\left(\dfrac{dy}{dx},\,\dfrac{d^2y}{dx^2}\right)\!$ at $x=0$ is:

If $G(x) = -\sqrt{25 - x^2}$ then $\lim_{x \to 1} \frac{G(x) - G(1)}{x - 1} =$ ______

If $2x = y^{1/5} + y^{-1/5}$ and $(x^2 - 1)\dfrac{d^2y}{dx^2} + \lambda x\dfrac{dy}{dx} + ky = 0$, then $\lambda + k$ is equal to

We have $\lim_{x \to 1} \dfrac{x^4 - 1}{x - 1} = \lim_{x \to k} \dfrac{x^3 - k^3}{x^2 - k^2}$. Find the value of $k$.

$\lim_{t \to 0} \lim_{x \to \infty} \cot\left(\frac{\pi(1 - t^2f(x) \cdot g(x))}{4}\right)$ is equal to

Let p = $\lim_{x \to 0^+} (1 + \tan^2 \sqrt{x})^{1/2x}$, then $\log p$ is equal to

lim x→0+ cos−1(x −[x]2) · sin−1(x −[x]2) x −x3 , where [x] denotes the greatest integer less than or equal to x, is:

If $\lim_{x \to 0}\left(1 + \int_0^{\sqrt{a^x-1}} (\sin(2\,\text{arc}\tan t))(1+t^2)^{\ln a}\,dt\right)^{\!\frac{1}{x}} = 5$, then find the value of $a$, where $a \in N$ and $a > 1,\; x > 0$.

If $\lim_{n \to \infty} \frac{\sqrt[n]{(2n)^n}}{n}$ is equals to $(a-1)e^b$, then

Find: $\lim_{b \to 0} \frac{r_{max} - r}{\sin b}$

Let $f(x) = \displaystyle\lim_{n \to \infty} \dfrac{x^{2n-1} + ax^3 + bx^2}{x^{2n} + 1}$ 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:

Let f(x) = 1−x(1+|1−x|) |1−x| cos 1 1−x for x ̸= 1. Find LHL and RHL at x = 1.

Consider a parabola $y = \frac{x^2}{4}$ and the point $F(0,1)$. Let $A_1(x_1, y_1), A_2(x_2, y_2), A_3(x_3, y_3), \ldots, A_N(x_n, y_n)$ are 'n' points on the parabola such that $x_k > 0$ and $\angle OFA_k = \frac{k\pi}{2n}$ $(k = 1,2,\ldots,n)$. If the value of $\lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^{n} FA_k = \frac{m}{\pi}$, then $m$ is ______.

Suppose $x_1 = \tan^{-1}2 > x_2 > x_3 > \ldots$ are the real numbers satisfying $\sin(x_{n+1} - x_n) + 2^{-(n+1)} \sin x_n \sin x_{n+1} = 0$ for all $n > 1$ and the sequence is convergent and $l = \lim_{n \to \infty} x_n$, the value of $4l$ is ______.

Let $H_n$ denotes the harmonic mean of $n$ positive integers $n+1, n+2, n+3, \ldots, n+n$. If $\lim_{n \to \infty} \left( \frac{H_n}{n} \right) = \frac{1}{k}$ then the value of $e^k$ is ______.

Let $f:\mathbb{R}\to(0,\infty)$ be a twice differentiable function such that $f(3)=18$, $f'(3)=0$ and $f''(3)=4$. Then $\displaystyle\lim_{x\to1}\left(\log_e\left(\dfrac{f(2+x)}{f(3)}\right)^{\frac{18}{(x-1)^2}}\right)$ is equal to:

The value of $\displaystyle\lim_{x\to0}\frac{\log_e(\sec(ex)\cdot\sec(e^2x)\cdots\sec(e^{10}x))}{e^2-e^{2\cos x}}$ is equal to

If $f(x) = x^4 \tan x^3 - x\ln(1 + x^2)$, then the value of $\dfrac{d^4 f(x)}{dx^4}$ at $x = 0$ is:

Let $ f(x) $ be defined as $ f(x) = kx + 2 $ for $ x \leq -1 $ and $ f(x) = 2x + 3 $ for $ x > -1 $. If $ f $ is continuous at $ x = -1 $, then $ k $ equals

Let $f(x) = \begin{cases}(x+1)(x+2), & x > 0 \\ a\sin x + b\cos x, & x \leq 0\end{cases}$. If f is differentiable at x = 0, find the value of a − b.

If $f(x)$ be such that $f(x) = \max(|3-x|, 3-x^3)$ then:

If $\lim_{x \to 0} \frac{x(1 + a\cos x) + b\sin x}{x^3} = 1$ then

If $e^y + xy = e$, the ordered pair $\left(\dfrac{dy}{dx}, \dfrac{d^2y}{dx^2}\right)$ at $x = 0$ is equal to:

If $f(x) = \begin{cases} \dfrac{\sin(p+1)x + \sin x}{x}, & x 0 \end{cases}$ is continuous at $x = 0$, then find the values of $p$ and $q$.

If $x = \sqrt{2^{\sec^{-1}t}}$ and $y = \sqrt{2^{\csc^{-1}t}}$ $(|t| \geq 1)$, then $\dfrac{dy}{dx}$ is equal to

Let $k = \lim_{x \to 0} \left( \frac{e^x(e^{nx}-1)}{e^x - 1} + e^x x^3 \right) = n$ and $f(x) = e^x + e^{2x} + e^{3x} + \cdots + e^{nx} + e^x \cdot x^3$. If $f'''(0) = 1^3 + 2^3 + 3^3 + \cdots + n^3 + 6 = 1302$, find the value of $k + n$.

Given $ x = 3\tan t,\ y = 3\sec t $, find $\left(\dfrac{d^2 y}{dx^2}\right)_{t=\pi/4}$.

If $\displaystyle\lim_{\alpha \to 0} \frac{e^{\cos(\alpha^n)} - e}{\alpha^m} = \frac{-e}{2}$ where $m$ and $n$ are positive integers greater than 1, then the value of $\dfrac{m}{n}$ is:

$\lim_{x \to \frac{\pi}{2}} \dfrac{\cot x - \cos x}{(\pi - 2x)^3}$ equals

The value of $\displaystyle\lim_{x \to 0}\left\lfloor (1-e^x)\frac{\sin x}{|x|}\right\rfloor$ equals: [Note: $[\,\cdot\,]$ denotes the greatest integer function.]

$f(x) = x^2 + 3, x \leq 1$$= 3x + a, x > 1$Is $f(x)$ neither continuous nor differentiable at $x = 1$?

Let $x^3 - 2x^2y^2 + 5x + y - 5 = 0$ and at $x = 1$, $y = 1$. Then $\dfrac{dy}{dx}$ at $y = 1$ is

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

$\lim_{x \to 2} [x]$ exists where $[x]$ denotes the integral part of $x$.State whether the statement is true or false.

If $f(x)$ is odd linear polynomial with $f(1) = 1$, then \[\lim_{x \to 0} \frac{2^{f(\tan x)} - 2^{f(\sin x)}}{x^2 f(\sin x)}\] is

\[\lim_{n \to \infty} \frac{e^n}{\left(1 + \dfrac{1}{n}\right)^{n^2}}\] equals

If $f(x)$ is a polynomial of least degree such that $\lim_{x \to 0}\left(1 + \dfrac{f(x) + x^2}{x^2}\right)^{1/x} = e^2$, then $f(2)$ is

$\lim_{x \to 0} \dfrac{e^{x^2} - \cos x}{\sin^2 x}$ is equal to

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

Let $f(2) = 4$ and $f'(2) = 4$. Then $\displaystyle\lim_{x \to 2} \frac{xf(2) - 2f(x)}{x - 2}$ is given by

If $f(x)$ is a real valued bijective function satisfying $f'(x) = \sin^2(\sin(x+1))$ and $f(0) = 3$, then the value of $(f^{-1})''(3)$ is equal to:

Let $f: R \to R$ be a function such that $|f(x)| \leq x^2$, for all $x \in R$. Then at $x = 0$, $f(x)$ is

If $ x = e^{y + e^{y + \cdots}} $, $ x > 0 $, then $ \dfrac{dy}{dx} $ is:

Given $f(t) = (|\lambda|e^{|t|} - \mu)\sin(2|t|)$. If $f(t)$ is differentiable at $t = 0$, then the set $S$ of all possible values of $(\lambda, \mu)$ is a subset of:

The value of $\displaystyle\lim_{x \to \infty} \frac{e^x\left[\left(2^{x^n}\right)^{1/e^x} - \left(e^{x^n}\right)^{1/e^x}\right]}{x^n}$ where $n$ is positive integer, is:

Given $f(x) = (x-1)^{\frac{1}{2-x}},\; x > 1,\; x \neq 2$ and $f(2) = k$. If $f$ is continuous at $x = 2$, find $k$.

If $\lim f(x)$ and $\lim g(x)$ exist then $\lim [f(x) \cdot g(x)]$ exist.

Let $ f(x) = \begin{cases} (x-1)\sin\left(\dfrac{1}{x-1}\right), & \text{if } x \neq 1 \\ 0, & \text{if } x = 1 \end{cases} $. Then which one of the following is true?

Continuity Questions (1086)