Limits Questions - JEE

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 $x$).

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

Limits Questions (1092)