Limits, Continuity and Differentiability

Let $x_0$ be the real number such that $e^{x_0} + x_0 = 0$. For a given real number $\alpha$, define

$$g(x) = \frac{3xe^x + 3x - \alpha e^x - \alpha x}{3(e^x + 1)}$$

for all real numbers $x$.

Then which one of the following statements is TRUE?

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

$$f(x) = \begin{cases} \dfrac{6x + \sin x}{2x + \sin x} & \text{if } x \neq 0,\\[2mm] \dfrac73 & \text{if } x = 0. \end{cases}$$

Then which of the following statements is (are) TRUE?

Let $f(x)$ be a continuously differentiable function on the interval $(0,\infty)$ such that $f(1) = 2$ and

$$\lim_{t\to x}\frac{t^{10}f(x) - x^{10}f(t)}{t^9 - x^9} = 1$$

for each $x > 0$. Then, for all $x > 0$, $f(x)$ is equal to

Let $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to \mathbb{R}$ be functions defined by

$$f(x) = \begin{cases} x|x|\sin\left(\dfrac{1}{x}\right), & x \neq 0, \\ 0, & x = 0, \end{cases} \quad\text{and}\quad g(x) = \begin{cases} 1 - 2x, & 0 \le x \le \dfrac{1}{2}, \\ 0, & \text{otherwise}. \end{cases}$$

Let $a, b, c, d \in \mathbb{R}$. Define the function $h : \mathbb{R} \to \mathbb{R}$ by

$$h(x) = a\,f(x) + b\left(g(x) + g\left(\frac{1}{2} - x\right)\right) + c\,(x - g(x)) + d\,g(x),\quad x \in \mathbb{R}.$$

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

The correct option is

Let $k \in \mathbb{R}$. If $\displaystyle\lim_{x\to 0+}\left(\sin(\sin kx) + \cos x + x\right)^{\frac{2}{x}} = e^6$, then the value of $k$ is

Let $S$ be the set of all $(\alpha, \beta) \in \mathbb{R}\times\mathbb{R}$ such that

$$\lim_{x\to\infty}\frac{\sin(x^2)(\log_e x)^{\alpha}\sin\left(\dfrac{1}{x^2}\right)}{x^{\alpha\beta}(\log_e(1+x))^{\beta}} = 0.$$

Then which of the following is (are) correct?

Let $S = (0,1)\cup(1,2)\cup(3,4)$ and $T = \{0, 1, 2, 3\}$. Then which of the following statements is(are) true?

Let $f : [0,1] \to [0,1]$ be the function defined by $f(x) = \dfrac{x^3}{3} - x^2 + \dfrac59x + \dfrac{17}{36}$. Consider the square region $S = [0,1]\times[0,1]$. Let $G = \{(x,y) \in S : y > f(x)\}$ be called the green region and $R = \{(x,y) \in S : y < f(x)\}$ be called the red region. Let $L_h = \{(x,h) \in S : x \in [0,1]\}$ be the horizontal line drawn at a height $h \in [0,1]$. Then which of the following statements is(are) true?

Let $f : (0,1) \to \mathbb{R}$ be the function defined as $f(x) = \sqrt n$ if $x \in \left[\dfrac{1}{n+1}, \dfrac1n\right)$ where $n \in \mathbb{N}$. Let $g : (0,1) \to \mathbb{R}$ be a function such that $\displaystyle\int_{x^2}^{x}\sqrt{\frac{1-t}{t}}\,dt < g(x) < 2\sqrt x$ for all $x \in (0,1)$. Then $\displaystyle\lim_{x\to0}f(x)g(x)$

Let $f : (0,1) \to \mathbb{R}$ be the function defined as $f(x) = [4x]\left(x - \dfrac14\right)^2\left(x - \dfrac12\right)$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements is(are) true?

Let $\alpha$ be a positive real number. Let $f : \mathbb{R} \to \mathbb{R}$ and $g : (\alpha, \infty) \to \mathbb{R}$ be the functions defined by
$$f(x) = \sin\left(\frac{\pi x}{12}\right) \quad\text{and}\quad g(x) = \frac{2\log_e\left(\sqrt x - \sqrt\alpha\right)}{\log_e\left(e^{\sqrt x} - e^{\sqrt\alpha}\right)}.$$
Then the value of $\displaystyle\lim_{x\to\alpha^+} f(g(x))$ is _____.

If
$$\beta = \lim_{x\to0} \frac{e^{x^3} - (1-x^3)^{\frac13} + \left((1-x^2)^{\frac12} - 1\right)\sin x}{x\sin^2 x},$$
then the value of $6\beta$ is _____.

For positive integer $n$, define
$$f(n) = n + \frac{16 + 5n - 3n^2}{4n + 3n^2} + \frac{32 + n - 3n^2}{8n + 3n^2} + \frac{48 - 3n - 3n^2}{12n + 3n^2} + \cdots + \frac{25n - 7n^2}{7n^2}.$$
Then, the value of $\displaystyle\lim_{n\to\infty} f(n)$ is equal to

The number of points where the function
$$f(x)=\begin{cases}|2x^2-3x-7| & \text{if } x\le-1\\ [4x^2-1] & \text{if } -1<x<1\\ |x+1|+|x-2| & \text{if } x\ge1,\end{cases}$$
$[t]$ denotes the greatest integer $\le t$, is discontinuous is ______.

Let
$$f(x)=\begin{cases}\dfrac{\sin(x-[x])}{x-[x]}, & x\in(-2,-1)\\[2mm] \max\{2x,3[|x|]\}, & |x|<1\\[1mm] 1, & \text{otherwise}\end{cases}$$
where $[t]$ denotes greatest integer $\le t$. If $m$ is the number of points where $f$ is not continuous and $n$ is the number of points where $f$ is not differentiable, then the ordered pair $(m,n)$ is:

Let $f(x)$ be a polynomial function such that $f(x)+f'(x)+f''(x)=x^5+64$. Then, the value of $\displaystyle\lim_{x\to1}\frac{f(x)}{x-1}$ is equal to :

$$\lim_{x\to\frac\pi2}\left(\tan^2x\left(\left(2\sin^2x+3\sin x+4\right)^{\frac12}-\left(\sin^2x+6\sin x+2\right)^{\frac12}\right)\right)$$
is equal to

Let $f(x)=\left[2x^2+1\right]$ and $g(x)=\begin{cases}2x-3, & x<0\\2x+3, & x\ge0\end{cases}$, where $[t]$ is the greatest integer $\le t$. Then, in the open interval $(-1,1)$, the number of points where $f\circ g$ is discontinuous is equal to ______.

$\displaystyle\lim_{x\to\frac1{\sqrt2}}\frac{\sin\left(\cos^{-1}x\right)-x}{1-\tan\left(\cos^{-1}x\right)}$ is equal to :

Let $f,g:\mathbb R\to\mathbb R$ be two real valued functions defined as $f(x)=\begin{cases}-|x+3|, & x<0\\e^x, & x\ge0\end{cases}$ and $g(x)=\begin{cases}x^2+k_1x, & x<0\\4x+k_2, & x\ge0\end{cases}$, where $k_1$ and $k_2$ are real constants. If $(g\circ f)$ is differentiable at $x=0$, then $(g\circ f)(-4)+(g\circ f)(4)$ is equal to :