Limits, Continuity & Differentiability Questions - JEE

For $ x > 1 $, if $ (2x)^{2y} = 4e^{2x-2y} $, then $ (1 + \log_e 2x)^2 \dfrac{dy}{dx} $ is equal to:

Let f, g: R → R be two functions defined by \[f(x) = \begin{cases} x\sin\left(\dfrac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}\] and $g(x) = xf(x)$.Statement 1: f is a continuous function at x = 0.Statement 2: g is a differentiable function at x = 0.

If $f(x) = x^n$, then the value of $f(1) - \dfrac{f'(1)}{1!} + \dfrac{f''(1)}{2!} - \dfrac{f'''(1)}{3!} + \cdots + \dfrac{(-1)^n f^{(n)}(1)}{n!}$ is:

The function $ f : R \sim \{0\} \to R $ given by \[ f(x) = \frac{1}{x} - \frac{2}{e^{2x} - 1} \] can be made continuous at $ x = 0 $ by defining $ f(0) $ as

If $f(x) = \begin{vmatrix} \sin x & \cos x & \tan x \\ x^3 & x^2 & x \\ 2x & 1 & 0\end{vmatrix}$, then which of the following are correct?

$f(x)=\begin{cases}x^2\cos(1/x) & x0\end{cases}$. Which are correct?

The value of $\lim_{n \to \infty} \left(\dfrac{n!}{n^n}\right)^{\frac{1}{n}}$ is equal to

If for $x \in \left(0, \dfrac{1}{4}\right)$, the derivative of $\tan^{-1}\left(\dfrac{6x\sqrt{x}}{1-9x^3}\right)$ is $\sqrt{x} \cdot g(x)$, then $g(x)$ equals

$g(x)=\begin{cases}x+b & x can be made differentiable at \(x=0$ if:

lim $x \to -\frac{1}{\sqrt{2}}$ \left( \sin(\cos^{-1} x) - x \right) is equal to

If $\lim_{x \to c} f(x) \cdot g(x)$ exists then both $\lim_{x \to c} f(x)$ and $\lim_{x \to c} g(x)$ exist.State whether this statement is true or false.

If $ \lim_{x \to 0} \dfrac{10 - \displaystyle\sum_{k=1}^{10}(\cos kx)}{x^2} = \dfrac{a}{b} $ where a and b are co-prime, then the value of $ (a + b) $ is equal to:

If $x^{2x} - 2x^x \cot y - 1 = 0$, then $\dfrac{dy}{dx}$ at $\left(1, \dfrac{\pi}{2}\right)$ is:

$\lim_{x \to 0} \dfrac{\sin(\pi\cos^2 x)}{x^2}$

The function $ f(x) = \left[ x^2 \left[ \dfrac{1}{x^2} \right] \right] $, $ x \neq 0 $ is ( [x] represents the greatest integer $ \leq x $)

$\lim_{x \to 0} \frac{2x^2 - \log(1+x)}{x^2}$ is equal to

Let $f$ be a differentiable function such that $\displaystyle\lim_{x \to 1} \frac{f(1+x^3-x)-f(x)}{\sin(x-1)} = \displaystyle\lim_{x \to 0} \frac{f(1-x)-f(1)}{x} + 10$, then $f'(1)$ is equal to:

The limit limx→0 3√ 1+x2−4√1−2x x+x2 is equal to:

$\lim_{x \to 0} (1 + \sin x)^{\cot x} =$ ______

If $\lim_{x \to 0} \dfrac{\log(3+x) - \log(3-x)}{x} = k$, the value of $k$ is

Given \[ f(x) = \begin{cases} 5, & \text{if } x \leq 1 \\ a + bx, & \text{if } 1 Then $f(x)$ is continuous for all $x$ for:

$\lim_{\theta \to \pi/4} \frac{\sqrt{2} - \cos\theta - \sin\theta}{(4\theta - \pi)^2}$ is equal to

limx→1/ √ 2 sin(cos−1 x)−x 1−tan(cos−1 x) is:

Find the value of a and b such that $\lim_{x \to 0} \dfrac{x(1 + a\cos x) - b\sin x}{x^3} = 1$. Then which one of the following could be true

Let f be a differentiable function satisfying the condition that f′(x) exists and the relation f(x + h) − f(x) = f(h) + x2h + xh2 holds. If f(0) = 0 and f′(0) = 1, find f(3).

If $f(x)$ be a continuous function in $[1, 3]$ and $f(x)$ takes rational values for all $x$ in the interval, and $f(2) = 10$ then $f(x) = 10$ for all $x \in [1, 3]$.State whether this statement is true or false.

$\lim_{x \to 2} \left( \dfrac{\sqrt{1 - \cos\{2(x-2)\}}}{x-2} \right)$

The value of $\lim_{n \to \infty} \frac{\log\left(1 + \sum_{K=1}^n \frac{1}{n}\right)}{e}$ is

Let $f : (-1, 1) \to R$ be a continuous function. If $\int_0^{\sin x} f(t)\,dt = \dfrac{\sqrt{3}}{2}\,x$, then $f\!\left(\dfrac{\sqrt{3}}{2}\right)$ is equal to

197. Let $g(x) = \dfrac{1}{f^{-1}(x)}$. Given the following data:$x$01234$f(x)$$-2$$-1$246$f'(x)$1/22/314/35/3The value of $g'(4)$ is:

limx→π/4 cot3 x−tan x cos(x+π/4) is equal to:

Let f be a differentiable function such that $ f'(x) = 7 - \dfrac{3}{4} \dfrac{f(x)}{x},\ (x > 0) $ and $ f(1) \neq 4 $. Then $ \lim_{x \to 0^+} x\, f\!\left(\dfrac{1}{x}\right) $:

Let $ f(x) $ be a function defined as \[ f(x) = \begin{cases} |1-x| & x f(x) is continuous and differentiable everywhere, find $ a + b + c + d $.

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 a positive integer, is:

Let $ f(x) = x + \sin x - [x + \sin x] + [x - \sin x] + [x] $ Find the number of points of discontinuity of $ f(x) $.

If $x^y = e^{x-y}$, then $\dfrac{dy}{dx}$ is

$\lim_{x \to \infty} \left(\frac{2^x+1}{2^x-1}\right)^{2^x} =$ ______

Given that $f(x) = x^a$, then $f(1) - \dfrac{f'(1)}{1!} + \dfrac{f''(1)}{2!} - \dfrac{f'''(1)}{3!} + \cdots + \dfrac{(-1)^n f^n(1)}{n!}$ equals:

limx→c f(x) does not exist when LHL ̸= RHL. Which of the following holds?

Let $f$ be a function defined by $y = f(x)$ where $x = 2t - |t|$ and $y = t^2 + t|t|$ for $t \in \mathbb{R}$, then:

If $\displaystyle\lim_{h \to 0} \frac{1}{h} f(1+h) = 5$, find $f'(1)$.

The number of points at which $[x^2]$ and $a^{[x^2]}$ are discontinuous and not differentiable in the interval \(1 is:

If $f(x) = \sqrt{\frac{x - \sin x}{x + \cos^2 x}}$ then $\lim_{x \to \infty} f(x)$ is

If limx→∞ x2+x+1 x+1 −ax −b = 4, then:

Let $ f(x) = 15 - |x - 10|;\; x \in R $. Then the set of all values of x, at which the function, $ g(x) = f(f(x)) $ is not differentiable, is:

If $f(x)$ is continuous and $f\!\left(\dfrac{9}{2}\right) = \dfrac{2}{9}$, then $\lim_{x \to 0} f\!\left(\dfrac{1 - \cos 3x}{x^2}\right)$ is equal to

If $a_1 = 1$ and $a_{n+1} = \dfrac{4 + 3a_n}{3 + 2a_n}$, $n \geq 1$ and if $\lim_{n \to \infty} a_n = a$, then the value of $a$ is

If $\lim_{x \to 0} \frac{a + bx\sin x + cx\cos x}{x^4} = 2$ then $a =$ ______, $b =$ ______, $c =$ ______

If $g > p > 0$ then $\lim_{x \to \infty} \frac{dx^p + ex^{p-2} + c}{dx^q + ex^{q-2} + b}$ is

$\lim_{x \to 0} \dfrac{(1-\cos 2x)(3+\cos x)}{x\tan 4x}$

Limits, Continuity & Differentiability Questions (1044)