Differentiability Questions - JEE

The value of $\lim_{n \to \infty} \left(\frac{n!}{n^n}\right)^{\frac{3n^3 + 4}{4n^4 - 1}}$, $n \in \mathbb{N}$ is equal to:

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

199. Let $f(x)$ be a function defined by $f(x) = (k - x^{10})^{1/10}$ where $k = 1025$ and $f'(2) = \dfrac{1}{f'(a)}$ where $a \in N$, then $a$ equals:

If $f(x)$ is twice differentiable and $f''(0) = p$ then $\lim_{x \to 0} \frac{2f(x) - 3f(2x) + f(4x)}{x^2}$ is

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:

Given $\lim_{x \to 0} \dfrac{\sin^2 x}{\sqrt{2} - \sqrt{1 + \cos x}}$Find the value of the limit.

Let $f(x) = \sqrt{x-2}$ and $g(x) = \sqrt{4-x^2}$, $x\in[-2,2]$. Which of the following are correct?

$\lim f(x) = \lim_{x \to 0} \frac{x[x]}{x^2}$

If $y = \sqrt{x+\sqrt{x+\sqrt{x+\cdots\infty}}}$, then $\dfrac{dy}{dx}$ at $x=2$ can be written as $p/q$ in lowest terms. Find $p+q$ (where answer is 36 from key — take $\dfrac{dy}{dx}=\dfrac{1}{2y-1}$ at $x=2$, $y=2$, so $dy/dx=1/3$, then $p+q=4$... revisiting: answer 36 = $\frac{1}{2y-1}$ evaluated at specific $x$).

The graph of function $y = f(x)$ has a unique tangent at $(e^a, 0)$ through which the graph passes. Then $\lim_{x \to e^a} \frac{\log(1 + 7f(x)) - \sin(f(x))}{3f(x)}$ equals

Let $ f(x) = \cot^{-1}\left(\text{sgn}\left(\dfrac{[x]}{2x - [x]}\right)\right) $:Statement-1: $ f(x) $ is discontinuous at $ x = 1 $.Statement-2: $ f(x) $ is non-differentiable at $ x = 1 $.Which of the following option is correct?[Note: $[k]$ denotes greatest integer function less than or equal to $k$.]

Given $\lim_{x \to 1^-} \dfrac{\sqrt{\pi} - \sqrt{2\sin^{-1}x}}{\sqrt{1-x}}$Find the value of the limit.

$\lim_{x \to 0} \left(\frac{1+5x^2}{1+3x^2}\right)^{\frac{1}{x^2}} =$ ______

Let $f: [-1,\,3] \to R$ be defined as \[f(x) = \begin{cases} |x| + [x], & -1 \leq x

Let a, b ∈ ℝ, (a ≠ 0). If the function f defined as\[f(x) = \begin{cases} \dfrac{2x^2}{a}, & 0 \le x is continuous in the interval $[0, \infty)$, then an ordered pair $(a, b)$ is

Let $f(x)$ be a continuous, periodic and bounded function with period 3 such that $\int_0^3 f(t)\,dt = 6$. Also $g'(x) = f(x)$, such that $g(0) = 0$. Find the value of $\lim_{x \to 0} xg\!\left(\dfrac{1}{x}\right)$.

If $f(0) = 1$, $f'(0) = -1$, $f(x) > 0$ for all $x$, then there exists a function $f(x)$ such that

$\lim_{x \to 1} f(x)$ exists if $f(x)$ is defined as follows:$f(x) = x^2, x \(= x, x = 1$$= x^2 + 2x, x > 1$State whether the statements are true or false.

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.]

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

If $f(x) = \begin{cases} \frac{\sin((p+1)x) + \sin x}{x}, & x 0 \end{cases}$ is continuous at $x = 0$, then the ordered pair $(p, q)$ is equal to

Let f(x) = xn, n being a non-negative integer. The value of n for which the equality f'(x + y) = f'(x) + f'(y) is valid for all x, y ≠ 0, is

Find the value of $ n $ if \[ \lim_{x \to 0} \frac{\ln(1 + \sin^3 x \cos^2 x)\cot(\ln^3(1+x))\tan^4 x}{\sin(\sqrt{x^2+2} - \sqrt{2}) \cdot \ln(1+x^2)} = \sqrt{n} \]

The set of points where $f(x) = \frac{x}{4 + |x|}$ is differentiable is

The derivative of an even function is an odd function.State whether the statement is true or false.

If $f(x) = \begin{cases} \frac{9e^{1/x} - e^{-1/x}}{x^2 e^{1/x}} & x \neq 0 \\ 0 & x = 0 \end{cases}$, then at $x = 0$, $f(x)$ is

If $y = \tan^{-1}\left(\dfrac{6x\sqrt{x}}{1-9x^3}\right)$, find $\dfrac{dy}{dx}$ in the form $\sqrt{x}\cdot g(x)$. Then $g(x)$ equals:

If f(x) = \begin{cases} \frac{1}{x} & x \neq 0 \\ 0 & x = 0 \end{cases}, then

If y is a function of x and log(x + y) = 2xy, then the value of y'(0) is

If a function $f(x)$ is defined as $f(x) = \begin{cases} 7x & x 1 \end{cases}$, then

Function $f(x) = \begin{cases} |x+1| & x (a) is continuous at all points in \(\mathbb{R}$(b) [option cut off in source text]

The minimum value of k (k ∈ ℤ) for which the equation $e^x = kx^2$ has exactly three real solutions, is

Evaluate: $\lim_{x \to 0} \dfrac{(1-\cos x) + (1-\cos 2x) + \cdots + (1-\cos 10x)}{x^2}$

If $\lim_{x \to 0} \frac{(a - n)nx - \tan x}{\sin nx} = 0$, $n \neq 0$, then $a$ is equal to:

The set of points where the function $f(x) = \frac{1}{(1+|x|)}\left(x-1\right)\sin\frac{x}{x-1}$, if $x \neq 1$ is differentiable, is

$\lim_{x \to 3} \dfrac{\sqrt{3x} - 3}{\sqrt{2x - 4} - \sqrt{2}}$ is equal to

Let f : (−1, 1) → R be a function defined by $ f(x) = \max\left\{-|x|, -\sqrt{1-x^2}\right\} $. If K be the set of all points at which f is not differentiable, then K has exactly:

If $f(9) = 9$, $f'(9) = 4$ then $\lim_{x \to 9} \frac{\sqrt{f(x)} - 3}{\sqrt{x} - 3} =$ ______

Let $f(x)$ be a polynomial of the second degree which is positive definite. If $g(x) = f(x) + f'(x) + f''(x)$ then for any real $x$

If $y = x + e^x$, then $\dfrac{d^2x}{dy^2}$ is:

If $x = y\ln(xy)$, then $\dfrac{dx}{dy}$ equals:

Given $f(1) = 5$, $f(2) = 8$, $f'(1) = 3$ and $f''(1) = 0$, find $f(3)$.

Evaluate: $\lim_{x \to 0^+} \dfrac{\displaystyle\int_0^{\tan^{-1} x} (\sin t^2)\, dt}{\dfrac{-x^3}{2}}$

Evaluate: $\lim_{x \to 0} \dfrac{1}{x} - \dfrac{2}{e^{2x} - 1}$

Find \lim_{x \to 0^-} f(x) and \lim_{x \to 1^-} f(x), where f(x) = \begin{cases} 2x + 3, & x \leq 0 \\ 3(x+1), & x > 0 \end{cases}

If $ f $ and $ g $ are differentiable functions in $[0, 1]$ satisfying $ f(0) = 2 = g(1) $, $ g(0) = 0 $ and $ f(1) = 6 $, then for some $ c \in [0, 1] $

The number of discontinuity of the greatest integer function $f(x) = [x]$, $x \in (-\frac{1}{2}, 100)$ is equal to

Let f(x) be a continuous function satisfying $\begin{cases} f(x) = x^2 - 6x + 8 & \text{for } -1 \le x \le a-1 \\ f(x) = f(x+a) \end{cases}$ where 'a' is a constant, then find the sum of all possible values of 'a'.

Let $f(x)$ be a continuous, periodic and bounded function with period 3 such that $\int_0^3 f(t)\,dt = 6$. Also $g'(x) = f(x)$, such that $g(0) = 0$. Find the value of $\lim_{x \to 0} x g\!\left(\dfrac{1}{x}\right)$.

Let $ f(x) = x|x| $ and $ g(x) = \sin x $.Statement-1: $ g \circ f $ is differentiable at $ x = 0 $ and its derivative is continuous at that point.Statement-2: $ g \circ f $ is twice differentiable at $ x = 0 $.

Differentiability Questions (1063)