Limits, Continuity & Differentiability Questions - JEE

Let $f:\mathbb{R}\to\mathbb{R}$ be defined as $$f(x)=\begin{cases}\dfrac{a-b\cos 2x}{x^2} & ,\; x<0\\x^2+cx+2 & ,\; 0\le x\le 1\\2x+1 & ,\; x>1\end{cases}$$ If $f$ is continuous everywhere in $\mathbb{R}$ and $m$ is the number of points where $f$ is NOT differentiable, then $m+a+b+c$ equals:

If function f(x) = \frac{\sqrt{1+x}-\sqrt{1-x}}{x}\cdot\frac{1}{3}\sqrt{1+x}\cdot\frac{1}{x}\cdot\sqrt{1}\cdot x is continuous function at x=0, then f(0) is equal to

If f(x) is a differentiable function for all x ∈ ℝ such that f(x) has fundamental period 2. f(x) = 0 has exactly two solutions in [0, 2], also f(0) ≥ 0. If minimum number of zeros of h(x) = f'(x)cos x + f(x)sin x in (0, 99π) is 120 − k, then k is …….

If $f(x) = x(\sqrt{x} - \sqrt{x+1})$ then

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

Suppose that f(0) = −3 and f′(x) ≤ 5 for all values of x. Then, the largest value which f(2) can assume is ________.

The value of $\lim_{x \to \infty} (\sqrt{a^2 x^2 + ax + 1} - \sqrt{a^2 x^2 + 1})$, where $a > 0$, is

If f is a differentiable function satisfying f−1(n) = 0, ∀n ≥ 1, n ∈ ℤ, then

The value of $\lim_{x \to 0} \frac{2\cot x}{x}$ is

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

The value of $\lim_{x \to 1^-} \frac{\int_1^x |t-1| dt}{\sin(x-1)}$ is

If f(x) = \begin{cases} 1+e^{1/x} & , x , then

The value of $\lim_{x \to 1} \{1 + x + [x-1] + [1-x]\}$ (where $[\cdot]$ denotes the greatest integer function) is

If $\lim_{x \to 1} \frac{1}{2^{x-1}}$ exists.

If $f(x) = \begin{cases} x^2\{x\}^2 + x\sin\{x\} & \text{for } x \neq 0 \\ 0 & \text{for } x = 0 \end{cases}$, where $\{x\}$ denotes the fractional part function, then

Let $f(x)$ be a function that is non-differentiable at $x = 0$ and $[2x]$ is discontinuous at $x = \frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}, 3$ (5 points). If $f(x)$ should be differentiable and continuous at $x = 2$, so $k = 3$, and given $f'(3^+) = f'(3^-)$ implies $2(3 - a) = 0$, so $a = 3$, and $f(3^+) = f(3^-)$ implies $(3 - a)^2 + b = 5$, so $b = 5$. Find the value of $a \cdot b \cdot k$.

If a function $f(x)$ is defined as \[f(x) = \begin{cases} -x & , x \leq 0 \\ x^2 & , 0 1 \end{cases}\] then

If f(x) = x/(1 + (log x)(log x)...), x ∈ [1, 3] is non-differentiable at x = k. Then, the value of [k²], is (where [ ] denotes greatest integer function)

If f(x) = \begin{cases} \frac{x^2+(a-2)x-2a}{x-2} & , x \neq 2 \\ 2 & , x = 2 \end{cases} is continuous at x=2, then a is equal to

Consider the piecewise defined function $f(x) = \begin{cases} -x & ; x 4 \end{cases}$. Choose the answer which best describes the continuity of this function.

The period of the function $f(x) = \sin(\sin(\pi x)) + e^{\{3x\}}$, where $\{.\}$ denotes the fractional part of $x$ is

Ex. 10 Let $f$ be a function such that $f(x + f(y)) = f(x) + y$, $\forall x, y \in \mathbb{R}$. Then find $f(0)$.If it is given that there exists a positive real $h$, such that $f(h) = h$ for $0 \leq h \leq H$, then find $f'(x)$.

$f(x) = 1 + |\sin x|$

Let $f(x) = \begin{cases} \left(\dfrac{2^x + 3^x + 5^x}{3}\right)^{3/x}, & x \neq 0 \\ k, & x = 0 \end{cases}$. If $f(x)$ is continuous then the value of $k$ is equal to:

If $\lim_{x \to 1} \frac{x + x^2 + x^3 + \ldots + x^n - n}{x - 1} = 820$, where $n \in \mathbb{N}$, then the value of $n$ is equal to ………

Let $f(x)=\begin{cases}\frac{a[x]+x-1}{[x]+x}, & x\neq 0\\\log_e a, & x=0\end{cases}$ where $a>0$. The function is

The number of points of discontinuity of $f(x) = [2x^3 - 5]$ in $[1, 2)$ is equal to(where $[\cdot]$ denotes the greatest integer function)

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

If $f(x) = (x-1)\sin\left(\dfrac{1}{x-1}\right)$ when $x \neq 1$ and $f(1) = 0$, then which of the following is true?

Let $f(x) = N(x) + O(x)$ where $N'(a)$ and $O'(a)$ are finite and definite. Which of the following statements is true?

If f(-10\sqrt{2}) = 2\sqrt{2}, then f'(-10\sqrt{2}) =

Which of the following is continuous everywhere in its domain but has at least one point where it is not differentiable?

Find the value of $\lim_{x \to 0} \frac{(1 - \cos 2x)(3 + \cos x)}{x \tan 4x}$

The number of points of discontinuity of $g(x) = \frac{1}{x^2 + 1 - f(x)}$ in $\left[\frac{-15}{2}, \frac{5}{2}\right]$ equals:

Let $f:\mathbb{R}\to(0,\infty)$ be a strictly increasing function such that $\displaystyle\lim_{x\to\infty}\dfrac{f(7x)}{f(x)}=1$. Then, the value of $\displaystyle\lim_{x\to\infty}\left[\dfrac{f(5x)}{f(x)}-1\right]$ is equal to

Since f(x) is continuous in $\left[0, \dfrac{\pi}{2}\right]$, find the value of $f\left(\dfrac{\pi}{4}\right) = \lim_{x \to \pi/4} f(x) = \lim_{x \to \pi/4} \left(\dfrac{1 - \tan x}{4x - \pi}\right)$. What is this limit equal to?

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

Limits, Continuity & Differentiability Questions (1044)