Limits, Continuity and Differentiability — JEE Main PYQs

The number of points in the interval $[2, 4]$, at which the function $f(x) = \left[x^2 - x - \frac{1}{2}\right]$, where $[\cdot]$ denotes the greatest integer function, is discontinuous, is __________.

The number of points, at which the function $f(x) = \max\{6x, 2 + 3x^2\} + |x - 1|\cos\left|x^2 - \frac{1}{4}\right|$, $x \in (-\pi, \pi)$, is not differentiable, is ______.

Let $f(x) = \begin{cases} e^{x-1} & , \ x < 0 \\ x^2 - 5x + 6 & , \ x \ge 0 \end{cases}$ and $g(x) = f(|x|) + |f(x)|$. If the number of points where $g$ is not continuous and is not differentiable are $\alpha$ and $\beta$ respectively, then $\alpha + \beta$ is equal to __________

The product of all possible values of $\alpha$, for which

$$\lim_{x \to 0} \left( \frac{1 - \cos(\alpha x)\cos\left( (\alpha+1)x \right)\cos\left( (\alpha+2)x \right)}{\sin^2\left( (\alpha+1)x \right)} \right) = 2, \text{ is:}$$

Let $f(x) = \displaystyle\lim_{y \to 0} \frac{(1 - \cos(xy))\tan(xy)}{y^3}$. Then the number of solutions of the equation $f(x) = \sin x$, $x \in \mathbf{R}$ is :

The value of $\lim\limits_{x \to 0}\left(\frac{x^2\sin^2 x}{x^2-\sin^2 x}\right)$ is:

Let $\lim_{x \to 2} \frac{(\tan(x - 2))\left(rx^2 + (p - 2)x - 2p\right)}{(x - 2)^2} = 5$ for some $r, p \in \mathbb{R}$. If the set of all possible values of $q$, such that the roots of the equation $rx^2 - px + q = 0$ lie in $(0, 2)$, be the interval $(\alpha, \beta]$, then $4(\alpha + \beta)$ equals :

Let $f(x) = \begin{cases} x^3 + 8\ ; & x < 0, \\ x^2 - 4\ ; & x \ge 0, \end{cases}$ and $g(x) = \begin{cases} (x - 8)^{1/3}\ ; & x < 0, \\ (x + 4)^{1/2}\ ; & x \ge 0. \end{cases}$ Then the number of points, where the function $g \circ f$ is discontinuous, is ________.

For the function $f(x) = e^{\sin|x|} - |x|$, $x \in \mathbb{R}$, consider the following statements :

Statement I : $f$ is differentiable for all $x \in \mathbb{R}$.

Statement II : $f$ is increasing in $\left(-\pi, -\frac{\pi}{2}\right)$.

In the light of the above statements, choose the correct answer from the options given below :

Let $f(x) = \begin{cases} \frac{1}{3}, & x \le \pi/2 \\ \frac{b(1 - \sin x)}{(\pi - 2x)^2}, & x > \pi/2 \end{cases}$. If $f$ is continuous at $x = \pi/2$, then the value of $\int_0^{3b-6} \left|x^2 + 2x - 3\right|\, dx$ is :

For $\alpha, \beta, \gamma \in \mathbf{R}$, if $\lim_{x \to 0} \frac{x^2 \sin\alpha x + (\gamma - 1)\mathrm{e}^{x^2}}{\sin 2x - \beta x} = 3$, then $\beta + \gamma - \alpha$ is equal to :

If $\displaystyle\lim_{x \to 0} \frac{\cos(2x)+a\cos(4x)-b}{x^4}$ is finite, then $(a+b)$ is equal to :

Let $f(x) = \begin{cases} (1+ax)^{1/x} & , \ x < 0 \\ 1+b & , \ x = 0 \\ \dfrac{(x+4)^{1/2} - 2}{(x+c)^{1/3} - 2} & , \ x > 0 \end{cases}$

be continuous at $x = 0$. Then $e^{a}bc$ is equal to:

If $\displaystyle\lim_{x \to 0}\left(\dfrac{\tan x}{x}\right)^{\frac{1}{x^2}} = p$, then $96\log_e p$ is equal to __________

If $\lim\limits_{x \to 1^{+}} \frac{(x-1)(6 + \lambda\cos(x-1)) + \mu\sin(1-x)}{(x-1)^3} = -1$, where $\lambda, \mu \in \mathbb{R}$, then $\lambda + \mu$ is equal to

Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function satisfying $f(0) = 1$ and $f(2x) - f(x) = x$ for all $x \in \mathbb{R}$. If $\lim\limits_{n \to \infty}\left\{f(x) - f\left(\frac{x}{2^n}\right)\right\} = G(x)$, then $\sum\limits_{r=1}^{10} G(r^2)$ is equal to

Let $m$ and $n$ be the number of points at which the function $f(x) = \max\{x, x^3, x^5, \ldots, x^{21}\}$, $x \in \mathbb{R}$, is not differentiable and not continuous, respectively. Then $m + n$ is equal to ________

Let $f$ be a differentiable function on $\mathbf{R}$ such that $f(2)=1, f'(2)=4$. Let $\lim\limits_{x\to 0}\left(f(2+x)\right)^{3/x}=\mathrm{e}^{\alpha}$. Then the number of times the curve $y=4x^3-4x^2-4(\alpha-7)x-\alpha$ meets $x$-axis is :

$$\lim_{x \to 0^+} \frac{\tan\left(5(x)^{\frac{1}{3}}\right) \log_e\left(1 + 3x^2\right)}{\left(\tan^{-1} 3\sqrt{x}\right)^2 \left(e^{5(x)^{\frac{4}{3}}} - 1\right)}$$ is equal to

The number of points of discontinuity of the function $f(x) = \left[\frac{x^2}{2}\right] - \left[\sqrt{x}\right]$, $x \in [0, 4]$, where $[\cdot]$ denotes the greatest integer function, is __________

For $t > -1$, let $\alpha_t$ and $\beta_t$ be the roots of the equation $\left(\left(t + 2\right)^{1/7} - 1\right)x^2 + \left(\left(t + 2\right)^{1/6} - 1\right)x + \left(\left(t + 2\right)^{1/21} - 1\right) = 0$. If $\lim\limits_{t \to -1^+} \alpha_t = a$ and $\lim\limits_{t \to -1^+} \beta_t = b$, then $72(a + b)^2$ is equal to _________.

If the function $f(x) = \dfrac{\tan(\tan x) - \sin(\sin x)}{\tan x - \sin x}$ is continuous at $x = 0$, then $f(0)$ is equal to _________.

Given below are two statements:

Statement I : $\lim\limits_{x \to 0}\left(\frac{\tan^{-1}x + \log_e\sqrt{\frac{1+x}{1-x}} - 2x}{x^5}\right) = \frac{2}{5}$

Statement II : $\lim\limits_{x \to 1}\left(x^{\frac{2}{1-x}}\right) = \frac{1}{e^2}$

In the light of the above statements, choose the correct answer from the options given below

Let $f(x)$ be a real differentiable function such that $f(0)=1$ and $f(x+y)=f(x)f'(y)+f'(x)f(y)$ for all $x, y \in \mathbb{R}$. Then $\sum_{n=1}^{100} \log_e f(n)$ is equal to :

If $\lim\limits_{x \to \infty} \left( \left( \dfrac{e}{1-e} \right) \left( \dfrac{1}{e} - \dfrac{x}{1+x} \right) \right)^{x} = \alpha$, then the value of $\dfrac{\log_e \alpha}{1 + \log_e \alpha}$ equals :

If the function

$$f(x) = \begin{cases} \frac{2}{x}\{\sin(k_1 + 1)x + \sin(k_2 - 1)x\}, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x}\log_e\left(\frac{2 + k_1 x}{2 + k_2 x}\right), & x > 0 \end{cases}$$

is continuous at $x = 0$, then $k_1^2 + k_2^2$ is equal to

$\lim\limits_{x \to \infty} \dfrac{(2x^2 - 3x + 5)(3x - 1)^{\frac{x}{2}}}{(3x^2 + 5x + 4)\sqrt{(3x + 2)^x}}$ is equal to :

$\lim_{x \to 0} \operatorname{cosec} x\left(\sqrt{2\cos^2 x + 3\cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)$ is:

Let $f: \mathbb{R}-\{0\} \to \mathbb{R}$ be a function such that $f(x)-6f\left(\frac{1}{x}\right)=\frac{35}{3x}-\frac{5}{2}$. If the $\lim_{x \to 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta$ ; $\alpha, \beta \in \mathbb{R}$, then $\alpha + 2\beta$ is equal to

Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + |x - 2|$, $-2 < x < 3$, is not continuous and not differentiable. Then $\mathrm{m + n}$ is equal to :

Let $f(x) = \begin{cases} 3x, & x < 0 \\ \min\{1 + x + [x], x + 2[x]\}, & 0 \le x \le 2 \\ 5, & x > 2, \end{cases}$ where $[.]$ denotes greatest integer function. If $\alpha$ and $\beta$ are the number of points, where $f$ is not continuous and is not differentiable, respectively, then $\alpha + \beta$ equals ____.

Let $f(x) = \lim_{n \to \infty} \sum_{r=0}^{n} \left(\frac{\tan\left(x/2^{r+1}\right) + \tan^3\left(x/2^{r+1}\right)}{1 - \tan^2\left(x/2^{r+1}\right)}\right)$. Then $\lim_{x \to 0} \frac{\mathrm{e}^x - \mathrm{e}^{f(x)}}{(x - f(x))}$ is equal to __________.

Let $f: \mathbf{R} \to \mathbf{R}$ be a function given by

$$f(x)=\begin{cases} \frac{1-\cos 2x}{x^{2}}, & x<0 \\ \alpha, & x=0, \\ \frac{\beta\sqrt{1-\cos x}}{x}, & x>0 \end{cases}$$

where $\alpha, \beta \in \mathbf{R}$. If $f$ is continuous at $x=0$, then $\alpha^{2}+\beta^{2}$ is equal to :

If $\lim\limits_{x \to 1} \frac{(5x+1)^{1/3}-(x+5)^{1/3}}{(2x+3)^{1/2}-(x+4)^{1/2}}=\frac{m\sqrt{5}}{n(2n)^{2/3}}$, where $\gcd(m, n)=1$, then $8m+12n$ is equal to ________.

If the function $$f(x) = \begin{cases} \dfrac{72^x - 9^x - 8^x + 1}{\sqrt{2} - \sqrt{1 + \cos x}} & , \ x \ne 0 \\ a \log_e 2 \log_e 3 & , \ x = 0 \end{cases}$$ is continuous at $x = 0$, then the value of $a^2$ is equal to

Let $f(x) = \int_0^x \left(t + \sin\left(1 - e^t\right)\right) dt, x \in \mathbb{R}$. Then, $\lim\limits_{x \to 0} \dfrac{f(x)}{x^3}$ is equal to

If the function $f(x) = \dfrac{\sin 3x + \alpha \sin x - \beta \cos 3x}{x^3}$, $x \in \mathbb{R}$, is continuous at $x = 0$, then $f(0)$ is equal to :

Let $f : [-1, 2] \rightarrow \mathbf{R}$ be given by $f(x) = 2x^2 + x + [x^2] - [x]$, where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is :

Let $a > 0$ be a root of the equation $2x^2 + x - 2 = 0$. If $\lim\limits_{x \to \frac{1}{a}} \dfrac{16\left(1 - \cos(2 + x - 2x^2)\right)}{(1 - ax)^2} = \alpha + \beta\sqrt{17}$, where $\alpha, \beta \in \mathbb{Z}$, then $\alpha + \beta$ is equal to ________.

Let $f : (-\infty, \infty) - \{0\} \to \mathbb{R}$ be a differentiable function such that $f'(1) = \lim\limits_{a \to \infty} a^2 f\left(\dfrac{1}{a}\right)$. Then $\lim\limits_{a \to \infty} \dfrac{a(a+1)}{2} \tan^{-1}\left(\dfrac{1}{a}\right) + a^2 - 2\log_e a$ is equal to

If $f(x) = \begin{cases} x^3 \sin\left(\dfrac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}$, then

$\lim\limits_{\mathrm{n}\to\infty} \dfrac{\left(1^2-1\right)(\mathrm{n}-1)+\left(2^2-2\right)(\mathrm{n}-2)+\cdots+\left((\mathrm{n}-1)^2-(\mathrm{n}-1)\right)\cdot 1}{\left(1^3+2^3+\cdots+\mathrm{n}^3\right)-\left(1^2+2^2+\cdots+\mathrm{n}^2\right)}$ is equal to :

Let $[t]$ denote the greatest integer less than or equal to $t$. Let $f:[0, \infty) \to \mathbf{R}$ be a function defined by $f(x)=\left[\dfrac{x}{2}+3\right]-\left[\sqrt{x}\right]$. Let $S$ be the set of all points in the interval $[0, 8]$ at which $f$ is not continuous. Then $\sum\limits_{\mathrm{a}\in S} \mathrm{a}$ is equal to ________.

The value of $\lim_{x\to 0} 2\left(\frac{1-\cos x\sqrt{\cos 2x}\,\sqrt[3]{\cos 3x}\,......\,\sqrt[10]{\cos 10x}}{x^2}\right)$ is __________.

For $a, b > 0$, let $f(x) = \begin{cases} \dfrac{\tan((a+1)x) + b\tan x}{x}, & x < 0 \\ 3, & x = 0 \\ \dfrac{\sqrt{ax + b^2x^2} - \sqrt{ax}}{b\sqrt{a}\, x\sqrt{x}}, & x > 0 \end{cases}$

be a continous function at $x = 0$. Then $\dfrac{b}{a}$ is equal to :

If $\alpha = \displaystyle\lim_{x \to 0^+} \left( \dfrac{\mathrm{e}^{\sqrt{\tan x}} - \mathrm{e}^{\sqrt{x}}}{\sqrt{\tan x} - \sqrt{x}} \right)$ and $\beta = \displaystyle\lim_{x \to 0} (1 + \sin x)^{\frac{1}{2}\cot x}$ are the roots of the quadratic equation $ax^2 + bx - \sqrt{\mathrm{e}} = 0$, then $12\log_e(a + b)$ is equal to ________.

Let $f:(0, \pi)\to\mathbf{R}$ be a function given by $f(x)=\begin{cases}\left(\frac{8}{7}\right)^{\frac{\tan 8x}{\tan 7x}}, & 0<x<\frac{\pi}{2}\\ \mathrm{a}-8, & x=\frac{\pi}{2}\\ \left(1+|\cot x|\right)^{\frac{\mathrm{b}}{\mathrm{a}}|\tan x|}, & \frac{\pi}{2}<x<\pi\end{cases}$

where $\mathrm{a}, \mathrm{b}\in\mathbf{Z}$. If $f$ is continuous at $x=\frac{\pi}{2}$, then $\mathrm{a}^2+\mathrm{b}^2$ is equal to ________.

Let $\lim\limits_{\mathrm{n}\to\infty}\left(\frac{\mathrm{n}}{\sqrt{\mathrm{n}^4+1}}-\frac{2\mathrm{n}}{(\mathrm{n}^2+1)\sqrt{\mathrm{n}^4+1}}+\frac{\mathrm{n}}{\sqrt{\mathrm{n}^4+16}}-\frac{8\mathrm{n}}{(\mathrm{n}^2+4)\sqrt{\mathrm{n}^4+16}}+\ldots+\frac{\mathrm{n}}{\sqrt{\mathrm{n}^4+\mathrm{n}^4}}-\frac{2\mathrm{n}\cdot\mathrm{n}^2}{(\mathrm{n}^2+\mathrm{n}^2)\sqrt{\mathrm{n}^4+\mathrm{n}^4}}\right)$ be $\frac{\pi}{\mathrm{k}}$, using only the principal values of the inverse trigonometric functions. Then $\mathrm{k}^2$ is equal to ________.

$\lim\limits_{x \to 0} \dfrac{e-(1+2x)^{\frac{1}{2x}}}{x}$ is equal to

$\lim\limits_{x \to \frac{\pi}{2}}\left(\dfrac{\int_{x^3}^{(\pi/2)^3}\left(\sin\left(2t^{1/3}\right)+\cos\left(t^{1/3}\right)\right)dt}{\left(x-\frac{\pi}{2}\right)^2}\right)$ is equal to

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 differential then $m + a + b + c$ equals :

Let $\{x\}$ denote the fractional part of $x$ and $f(x) = \dfrac{\cos^{-1}\left(1 - \{x\}^2\right)\sin^{-1}(1 - \{x\})}{\{x\} - \{x\}^3}$, $x \neq 0$. If L and R respectively denotes the left hand limit and the right hand limit of $f(x)$ at $x = 0$, then $\dfrac{32}{\pi^2}\left(\mathrm{L}^2 + \mathrm{R}^2\right)$ is equal to ________.

Let $f(x)=|2x^2+5|x|-3|$, $x \in \mathbf{R}$. If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m+n$ is equal to :

Let $f(x)=\begin{cases} x-1, & x \text{ is even,} \\ 2x, & x \text{ is odd,} \end{cases} \; x \in \mathbf{N}$. If for some $a \in \mathbf{N}$, $f(f(f(a)))=21$, then $\lim_{x \to a^-}\left\{\frac{|x|^3}{a}-\left[\frac{x}{a}\right]\right\}$, where $[t]$ denotes the greatest integer less than or equal to $t$, is equal to :

Consider the function.

$$f(x) = \begin{cases} \dfrac{a(7x - 12 - x^2)}{b|x^2 - 7x + 12|} & , \ x < 3 \\ 2^{\frac{\sin(x-3)}{x - [x]}} & , \ x > 3 \\ b & , \ x = 3 , \end{cases}$$

where $[x]$ denotes the greatest integer less than or equal to $x$. If S denotes the set of all ordered pairs (a, b) such that $f(x)$ is continuous at $x = 3$, then the number of elements in S is :

If $a = \lim\limits_{x \to 0} \dfrac{\sqrt{1 + \sqrt{1 + x^4}} - \sqrt{2}}{x^4}$ and $b = \lim\limits_{x \to 0} \dfrac{\sin^2 x}{\sqrt{2} - \sqrt{1 + \cos x}}$, then the value of $ab^3$ is :

$\lim\limits_{x \to \frac{\pi}{2}}\left(\dfrac{1}{\left(x-\frac{\pi}{2}\right)^2}\int\limits_{x^3}^{\left(\frac{\pi}{2}\right)^3} \cos\left(t^{\frac{1}{3}}\right)dt\right)$ is equal to

Let $f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\to \mathbf{R}$ be a differentiable function such that $f(0)=\frac{1}{2}$. If the $\lim\limits_{x\to 0} \frac{x\int_0^x f(t)\,\mathrm{d}t}{e^{x^2}-1}=\alpha$, then $8\alpha^2$ is equal to :

Let $g:\mathbf{R}\to\mathbf{R}$ be a non constant twice differentiable function such that $g'\left(\frac{1}{2}\right)=g'\left(\frac{3}{2}\right)$. If a real valued function $f$ is defined as $f(x)=\frac{1}{2}[g(x)+g(2-x)]$, then

If the function $$f(x)=\begin{cases} \frac{1}{|x|}, & |x|\geq 2 \\ ax^2+2b, & |x|<2 \end{cases}$$ is differentiable on $\mathbf{R}$, then $48(a+b)$ is equal to ________.

Let $a$ and $b$ be real constants such that the function $f$ defined by $f(x)=\begin{cases} x^2+3x+a & ,\ x \leq 1 \\ bx+2 & ,\ x > 1 \end{cases}$ be differentiable on $\mathbb{R}$. Then, the value of $\int_{-2}^{2} f(x)\,dx$ equals

$\lim_{x \to 0} \frac{e^{2|\sin x|}-2|\sin x|-1}{x^2}$

Let $g(x)$ be a linear function and $f(x)=\begin{cases} g(x) & , x \leq 0 \\ \left(\frac{1+x}{2+x}\right)^{\frac{1}{x}} & , x>0 \end{cases}$, is continuous at $x=0$. If $f'(1)=f(-1)$, then the value $g(3)$ is