Limits Questions - JEE

If f(x) = sin $\frac{[x]}{x^2}$ for 2 ≤ x ≤ 3 and [x] denotes the greatest integer less than or equal to x, then f'(π/3) is equal to

Let $f(x) = \frac{\sqrt{2 + \cos x}}{8x - 4\sqrt{x}}$, $g(x) = \frac{2\cos x - \sin 2x}{e^{2x} - 1}$ (where $0 + 2x$ in denominator), and $h(x) = \begin{cases} f(x) & \text{for } x \pi/2 \end{cases}$. Which of the following does not hold?

The value of $\lim_{n \to \infty} \frac{1}{n}\left([1^2 x^7 + 1^2] + [2^2 x^7 + 2^2] + \ldots + [n^2 x^7 + n^2]\right)$ (where $[\cdot]$ denotes the greatest integer function) is

If $f(x) = \begin{cases} \frac{(4x-1)^3}{x^2} \sin\left(\log\left(1 + \frac{x^2}{a}\right)\right) \left(\log\frac{x}{3}\right)^n, & x \neq 0 \\ \frac{9(\log 4)^3}{2}, & x = 0 \end{cases}$ is a continuous function at x = 0, then the value of a is equal to

Find $\lim_{b \to 0} \frac{r_{max} - r}{\sin b}$

Find the value of $\lim_{\theta \to 0} \frac{\tan(\pi \cos^2 \theta)}{\sin(2\pi \sin^2 \theta)}$

Let f and g be differentiable functions satisfying g'(a) = 2, g(a) = b and f∘g = I (Identity function). Then, f'(b) is equal to

If $x = f(t), y = y(t)$, then $\frac{d^2y}{dx^2} =$

The value of $\lim_{n \to \infty} \left(\frac{1}{na} + \frac{1}{na+1} + \frac{1}{na+2} + \ldots + \frac{1}{nb}\right)$ is

Suppose $ f(x) $ is differentiable at $ x = 1 $ and $ \lim_{h \to 0} \frac{1}{h} f(1+h) = 5 $, then $ f'(1) $ equals

lim $x \to 1$ \frac{$\sqrt{\pi}$ - $\sqrt{2}$ $\sin^{-1} x$}{$\sqrt{1 - x}$} equal to :

If $\sin y = x \sin(\alpha + y)$, then $\dfrac{dy}{dx}$ is

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

The value of $\lim_{n \to \infty} \cos^2\left(\pi\left(3n^3 + n^2 + 2n\right)\right)$ (where $n \in \mathbb{N}$):

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

$\lim_{x\to\infty}\dfrac{3}{x}\left[\dfrac{x}{4}\right]=\dfrac{p}{q}$ (where $[\cdot]$ denotes GIF), then $p+q$ (where $p,q$ are relative prime) is

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

Limits Questions (1092)