Limits, Continuity & Differentiability Questions - JEE

●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:

$\lim_{x \to 0} \frac{\int_0^x \frac{e^{\sin(tx)}}{x} dt}{x}$ equals to:

Evaluate: $\lim_{x \to \pi} \dfrac{\sqrt{2 + \cos x} - 1}{(\pi - x)^2}$. If this limit equals $k$, find $k$.

If $\lim_{x \to 0} \frac{a\sin x - bx + cx^2 + x^3}{2x^3\ln(1+x) - 2x^3 + x^4}$ exists and is equal to $l$ then $a + b + c + l = $ ______.

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

Let $ k $ be a non-zero real number. If \[ f(x) = \begin{cases} \dfrac{(e^x - 1)^2}{\sin\!\left(\dfrac{x}{k}\right)\log\!\left(1 + \dfrac{x}{4}\right)}, & x \neq 0 \\ 12, & x = 0 \end{cases} \] is a continuous function, then the value of $ k $ is

If $f(x) = x^2 - x + 5$, $x > \dfrac{1}{2}$ and $g(x)$ is its inverse function, then $g'(7)$ equals

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

Consider the function $f(x) = \begin{cases} \sqrt{x^2 - 9} & \text{for some condition} \end{cases}$, then

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:

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function having $f(2) = 6$, $f'(2) = \left(\dfrac{1}{48}\right)$. Then $\lim_{x \to 2} \int_6^{f(x)} \dfrac{4t^3}{x-2}\, dt$ equals

Evaluate: $\displaystyle\lim_{x \to 0} \dfrac{x\cot(4x)}{\sin^2 x \cot^2(2x)}$

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 $x$).

Limits, Continuity & Differentiability Questions (1044)