Limits, Continuity and Differentiability

Let $\mathbb R$ denote the set of all real numbers. Let $f:\mathbb R\to\mathbb R$ be an arbitrary function and let $g:\mathbb R\to\mathbb R$ be the function defined by
$$g(x)=xf(x),\quad\text{for all }x\in\mathbb R.$$
Then which of the following statements is (are) TRUE ?

Consider the function $f:\left(-\dfrac\pi2,\dfrac\pi2\right)\to(-\infty,\infty)$ defined by
$$f(x)=(|x|+|x-1|)\sin x+[x\sin x],$$
where $[x\sin x]$ is the greatest integer less than or equal to $x\sin x$.

Let $\alpha$ be the total number of points in the interval $\left(-\dfrac\pi2,\dfrac\pi2\right)$ at which $f$ is NOT continuous, and let $\beta$ be the total number of points in the interval $\left(-\dfrac\pi2,\dfrac\pi2\right)$ at which $f$ is NOT differentiable.

Then the value of $\alpha+\beta$ is ________.

For a real number $\alpha$, let $[\alpha]$ denote the greatest integer less than or equal to $\alpha$. For a finite set $S$, let $|S|$ denote the number of elements in the set $S$.

Consider the functions $f:(-3,3)\to(-\infty,\infty)$ and $g:(-3,3)\to(-\infty,\infty)$ defined by
$$f(x)=\left[x^3\right]\log_e\left(1+\sin^2\left(\pi(x-[x])\right)\right)$$
and
$$g(x)=x^3\sin^2\left(\pi\log_e(1+x-[x])\right).$$
Let
$$A=\{x\in(-3,3):f\text{ is discontinuous at }x\}$$
and
$$B=\{x\in(-3,3):g\text{ is discontinuous at }x\}.$$
Then the value of $|A|+2|B|-|A\cap B|$ is ________.

If $\displaystyle\lim_{x\to2}\frac{\sin\left(x^3-5x^2+ax+b\right)}{\left(\sqrt{x-1}-1\right)\log_e(x-1)}=m$, then $a+b+m$ is equal to :

The number of critical points of the function $f(x)=\begin{cases}\left|\dfrac{\sin x}x\right|, & x\ne0\\[4pt]1, & x=0\end{cases}$ in the interval $(-2\pi,2\pi)$ is equal to :

Let $f(x)$ be a polynomial of degree 5, and have extrema at $x=1$ and $x=-1$. If $\displaystyle\lim_{x\to0}\left(\frac{f(x)}{x^3}\right)=-5$, then $f(2)-f(-2)$ is equal to :

The number of points in the interval $[2,4]$, at which the function $f(x)=\left[x^2-x-\dfrac12\right]$, where $[\cdot]$ denotes the greatest integer function, is discontinuous, is ________.

The number of points, at which the function $f(x)=\max\{6x,\,2+3x^2\}+|x-1|\cos\left|x^2-\dfrac14\right|$, $x\in(-\pi,\pi)$, is not differentiable, is ________.

Let $f(x)=\begin{cases}e^{x-1}, & x<0\\ x^2-5x+6, & x\ge0\end{cases}$ 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
$$\lim_{x\to0}\left(\frac{1-\cos(\alpha x)\cos((\alpha+1)x)\cos((\alpha+2)x)}{\sin^2((\alpha+1)x)}\right)=2,$$
is:

Let $f(x)=\displaystyle\lim_{y\to0}\frac{(1-\cos(xy))\tan(xy)}{y^3}$. Then the number of solutions of the equation $f(x)=\sin x$, $x\in\mathbf R$ is :

The value of $\displaystyle\lim_{x\to0}\left(\frac{x^2\sin^2x}{x^2-\sin^2x}\right)$ is:

Let $\displaystyle\lim_{x\to2}\frac{\left(\tan(x-2)\right)\left(rx^2+(p-2)x-2p\right)}{(x-2)^2}=5$ for some $r,p\in\mathbf R$. 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 $f(x)=\begin{cases}x^3+8; & x<0,\\ x^2-4; & x\ge0,\end{cases}$ and $g(x)=\begin{cases}(x-8)^{1/3}; & x<0,\\ (x+4)^{1/2}; & x\ge0.\end{cases}$ Then the number of points, where the function $gof$ is discontinuous, is ________.

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

Statement I : $f$ is differentiable for all $x\in\mathbf R$.

Statement II : $f$ is increasing in $\left(-\pi,-\dfrac\pi2\right)$.

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

Let $f(x)=\begin{cases}\dfrac13, & x\le\dfrac\pi2\\[2mm] \dfrac{b(1-\sin x)}{(\pi-2x)^2}, & x>\dfrac\pi2\end{cases}$. If $f$ is continuous at $x=\pi/2$, then the value of $\displaystyle\int_0^{3b-6}\left|x^2+2x-3\right|dx$ is :

Let $\mathbb{R}$ denote the set of all real numbers. Define the function $f : \mathbb{R} \to \mathbb{R}$ by

$$f(x) = \begin{cases} 2 - 2x^2 - x^2\sin\dfrac{1}{x} & \text{if } x \neq 0,\\ 2 & \text{if } x = 0. \end{cases}$$

Then which one of the following statements is TRUE?

Let $\alpha$ and $\beta$ be the real numbers such that

$$\lim_{x\to0}\frac{1}{x^3}\left(\frac{\alpha}{2}\int_0^x\frac{1}{1-t^2}\,dt + \beta x\cos x\right) = 2.$$

Then the value of $\alpha + \beta$ is _____.

For all $x > 0$, let $y_1(x)$, $y_2(x)$, and $y_3(x)$ be the functions satisfying

$$\begin{aligned} &\frac{dy_1}{dx} - (\sin x)^2\,y_1 = 0, & y_1(1) &= 5,\\ &\frac{dy_2}{dx} - (\cos x)^2\,y_2 = 0, & y_2(1) &= \frac13,\\ &\frac{dy_3}{dx} - \left(\frac{2 - x^3}{x^3}\right)y_3 = 0, & y_3(1) &= \frac{3}{5e}, \end{aligned}$$

respectively. Then

$$\lim_{x\to0^+}\frac{y_1(x)\,y_2(x)\,y_3(x) + 2x}{e^{3x}\sin x}$$

is equal to _____.

Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let $[x]$ denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number.

Match each entry in List-I to the correct entry in List-II and choose the correct option.