Limits, Continuity and Differentiability

$\displaystyle\lim_{x\to0}\frac{\cos(\sin x)-\cos x}{x^4}$ is equal to :

Let $f(x)=\min\{1,1+x\sin x\}$, $0\le x\le2\pi$. If $m$ is the number of points, where $f$ is not differentiable and $n$ is the number of points, where $f$ is not continuous, then the ordered pair $(m,n)$ is equal to

Let $a$ be an integer such that $\displaystyle\lim_{x\to7}\frac{18-[1-x]}{[x-3a]}$ exists, where $[t]$ is greatest integer $\le t$. Then $a$ is equal to :

Let $[t]$ denote the greatest integer $\le t$ and $\{t\}$ denote the fractional part of $t$. The integral value of $\alpha$ for which the left hand limit of the function
$$f(x)=[1+x]+\frac{\alpha^{2[x]+\{x\}}+[x]-1}{2[x]+\{x\}}$$
at $x=0$ is equal to $\alpha-\dfrac43$, is ______.

Let $f:\mathbb R\to\mathbb R$ be defined as
$$f(x)=\begin{cases}\left[e^x\right], & x<0\\ ae^x+[x-1], & 0\le x<1\\ b+[\sin(\pi x)], & 1\le x<2\\ \left[e^{-x}\right]-c, & x\ge2\end{cases}$$
where $a,b,c\in\mathbb R$ and $[t]$ denotes greatest integer less than or equal to $t$. Then, which of the following statements is true ?

Let $f,g:\mathbf R\to\mathbf R$ be functions defined by
$$f(x)=\begin{cases}[x], & x<0\\ |1-x|, & x\ge0\end{cases}\quad\text{and}\quad g(x)=\begin{cases}e^x-x, & x<0\\ (x-1)^2-1, & x\ge0\end{cases}$$
where $[x]$ denote the greatest integer less than or equal to $x$. Then, the function $f\circ g$ is discontinuous at exactly :

Let $f:\mathbf R\to\mathbf R$ be a differentiable function such that $f\left(\dfrac\pi4\right)=\sqrt2$, $f\left(\dfrac\pi2\right)=0$ and $f'\left(\dfrac\pi2\right)=1$ and let $g(x)=\displaystyle\int_x^{\pi/4}\left(f'(t)\sec t+\tan t\sec t\,f(t)\right)dt$ for $x\in\left[\dfrac\pi4,\dfrac\pi2\right)$. Then $\displaystyle\lim_{x\to\left(\frac\pi2\right)^-}g(x)$ is equal to :

If $\displaystyle\lim_{x\to1}\frac{\sin\left(3x^2-4x+1\right)-x^2+1}{2x^3-7x^2+ax+b}=-2$, then the value of $(a-b)$ is equal to ______.

Let $f:\mathbf R\to\mathbf R$ be a function defined by :
$$f(x)=\begin{cases}\displaystyle\max_{t\le x}\left\{t^3-3t\right\}, & x\le2\\ x^2+2x-6, & 2<x<3\\ [x-3]+9, & 3\le x\le5\\ 2x+1, & x>5\end{cases}$$
where $[t]$ is the greatest integer less than or equal to $t$. Let $m$ be the number of points where $f$ is not differentiable and $I=\displaystyle\int_{-2}^2f(x)\,dx$. Then the ordered pair $(m,I)$ is equal to :

The value of $\displaystyle\lim_{x\to1}\frac{\left(x^2-1\right)\sin^2(\pi x)}{x^4-2x^3+2x-1}$ is equal to:

Suppose $\displaystyle\lim_{x\to0}\frac{F(x)}{x^3}$ exists and is equal to L, where
$$F(x)=\begin{vmatrix}a+\sin\frac x2&-b\cos x&0\\-b\cos x&0&a+\sin\frac x2\\0&a+\sin\frac x2&-b\cos x\end{vmatrix}.$$
Then, $-112L$ is equal to ______.

If $\displaystyle\lim_{n\to\infty}\left(\sqrt{n^2-n-1}+n\alpha+\beta\right)=0$, then $8(\alpha+\beta)$ is equal to :

Let $f(x)=\begin{cases}\left|4x^2-8x+5\right|, & \text{if }8x^2-6x+1\ge0\\ \left[4x^2-8x+5\right], & \text{if }8x^2-6x+1<0,\end{cases}$ where $[\alpha]$ denotes the greatest integer less than or equal to $\alpha$. Then the number of points in $\mathbf R$ where $f$ is not differentiable is ______.

If $\displaystyle\lim_{n\to\infty}\frac{(n+1)^{k-1}}{n^{k+1}}\left[(nk+1)+(nk+2)+\dots+(nk+n)\right]=33\cdot\lim_{n\to\infty}\frac1{n^{k+1}}\cdot\left[1^k+2^k+3^k+\dots+n^k\right]$, then the integral value of k is equal to ______.

$\displaystyle\lim_{x\to\frac\pi4}\frac{8\sqrt2-(\cos x+\sin x)^7}{\sqrt2-\sqrt2\sin2x}$ is equal to

Let $f:\mathbf R\to\mathbf R$ be a continuous function such that $f(3x)-f(x)=x$. If $f(8)=7$, then $f(14)$ is equal to :

If the function $f(x)=\begin{cases}\dfrac{\log_e\left(1-x+x^2\right)+\log_e\left(1+x+x^2\right)}{\sec x-\cos x}, & x\in\left(\dfrac{-\pi}2,\dfrac\pi2\right)-\{0\}\\ k, & x=0\end{cases}$ is continuous at $x=0$, then k is equal to :

If $f(x)=\begin{cases}x+a, & x\le0\\|x-4|, & x>0\end{cases}$ and $g(x)=\begin{cases}x+1, & x<0\\(x-4)^2+b, & x\ge0\end{cases}$ are continuous on $\mathbf R$, then $(g\circ f)(2)+(f\circ g)(-2)$ is equal to :

Let $\beta=\displaystyle\lim_{x\to0}\frac{\alpha x-\left(e^{3x}-1\right)}{\alpha x\left(e^{3x}-1\right)}$ for some $\alpha\in\mathbf R$. Then the value of $\alpha+\beta$ is :

Let $f:\mathbf R\to\mathbf R$ be a function defined as

$$f(x)=a\sin\left(\frac{\pi[x]}2\right)+[2-x],\ a\in\mathbf R,$$

where $[t]$ is the greatest integer less than or equal to $t$. If $\displaystyle\lim_{x\to-1}f(x)$ exists, then the value of $\displaystyle\int_0^4f(x)\,dx$ is equal to