Limits, Continuity and Differentiability

Let $a,b\in\mathbb R$ and $f:\mathbb R\to\mathbb R$ be defined by $f(x)=a\cos(|x^3-x|)+b|x|\sin(|x^3+x|)$. Then $f$ is

Let $f:\mathbb R\to(0,\infty)$ and $g:\mathbb R\to\mathbb R$ be twice differentiable functions such that $f''$ and $g''$ are continuous functions on $\mathbb R$. Suppose $f'(2)=g(2)=0$, $f''(2)\ne0$ and $g'(2)\ne0$. If $\displaystyle\lim_{x\to2}\frac{f(x)g(x)}{f'(x)g'(x)}=1$, then

Let $f:\left[-\frac12,2\right]\to\mathbb R$ and $g:\left[-\frac12,2\right]\to\mathbb R$ be functions defined by $f(x)=[x^2-3]$ and $g(x)=|x|f(x)+|4x-7|f(x)$, where $[y]$ denotes the greatest integer less than or equal to $y$ for $y\in\mathbb R$. Then

Let $p=\displaystyle\lim_{x\to0+}\left(1+\tan^2\sqrt x\right)^{\frac1{2x}}$ then log p is equal to :

For $x\in\mathbb R$, $f(x)=|\log2-\sin x|$ and $g(x)=f(f(x))$, then :

$\displaystyle\lim_{n\to\infty}\left(\frac{(n+1)(n+2)\dots3n}{n^{2n}}\right)^{1/n}$ is equal to :

If $\displaystyle\lim_{x\to\infty}\left(1+\frac ax-\frac4{x^2}\right)^{2x}=e^3$, then 'a' is equal to :

If the function $f(x)=\begin{cases}-x, & x<1\\ a+\cos^{-1}(x+b), & 1\le x\le2\end{cases}$ is differentiable at x = 1, then $\dfrac ab$ is equal to :

If f(x) is a differentiable function in the interval (0, ∞) such that f(1) = 1 and $\displaystyle\lim_{t\to x}\frac{t^2f(x)-x^2f(t)}{t-x}=1$, for each x > 0, then $f\left(\dfrac32\right)$ is equal to :

$\displaystyle\lim_{x\to0}\frac{(1-\cos2x)^2}{2x\tan x-x\tan2x}$ is :

Let a, b ∈ R, (a ≠ 0). If the function f defined as
$$f(x)=\begin{cases}\dfrac{2x^2}a, & 0\le x<1\\ a, & 1\le x<\sqrt2\\ \dfrac{2b^2-4b}{x^3}, & \sqrt2\le x<\infty\end{cases}$$
is continuous in the interval [0, ∞), then an ordered pair (a, b) is :

Let $g:\mathbb R\to\mathbb R$ be a differentiable function with $g(0)=0$, $g'(0)=0$ and $g'(1)\ne0$. Let
$$f(x)=\begin{cases}\dfrac{x}{|x|}g(x),&x\ne0\\0,&x=0\end{cases}$$
and $h(x)=e^{|x|}$ for all $x\in\mathbb R$. Let $(f\circ h)(x)$ denote $f(h(x))$ and $(h\circ f)(x)$ denote $h(f(x))$. Then which of the following is (are) true?

Let $f(x)=\sin\left(\dfrac\pi6\sin\left(\dfrac\pi2\sin x\right)\right)$ for all $x\in\mathbb R$ and $g(x)=\dfrac\pi2\sin x$ for all $x\in\mathbb R$. Let $(f\circ g)(x)$ denote $f(g(x))$ and $(g\circ f)(x)$ denote $g(f(x))$. Then which of the following is (are) true?

Let $m$ and $n$ be two positive integers greater than 1. If
$$\lim_{\alpha\to0}\left(\frac{e^{\cos(\alpha^n)}-e}{\alpha^m}\right)=-\left(\frac e2\right)$$
then the value of $\dfrac mn$ is

Let $f:\mathbb R\to\mathbb R$ be a continuous odd function, which vanishes exactly at one point and $f(1)=\dfrac12$. Suppose that $F(x)=\displaystyle\int_{-1}^xf(t)\,dt$ for all $x\in[-1,2]$ and $G(x)=\displaystyle\int_{-1}^xt|f(f(t))|\,dt$ for all $x\in[-1,2]$. If $\displaystyle\lim_{x\to1}\frac{F(x)}{G(x)}=\frac1{14}$, then the value of $f\left(\dfrac12\right)$ is

$\displaystyle\lim_{x\to0}\frac{(1-\cos2x)(3+\cos x)}{x\tan4x}$ is equal to :

If the function.
$$g(x)=\begin{cases}k\sqrt{x+1},&0\le x\le3\\mx+2,&3<x\le5\end{cases}$$
is differentiable, then the value of $k+m$ is :

Let $f(x)$ be a polynomial of degree four having extreme values at $x=1$ and $x=2$. If $\displaystyle\lim_{x\to0}\left[1+\frac{f(x)}{x^2}\right]=3$, then $f(2)$ is equal to :

$\displaystyle\lim_{x\to0}\frac{e^{x^2}-\cos x}{\sin^2x}$ is equal to :

Let k be a non-zero real number. If
$$f(x)=\begin{cases}\dfrac{(e^x-1)^2}{\sin\left(\dfrac xk\right)\log\left(1+\dfrac x4\right)},&x\ne0\\12,&x=0\end{cases}$$
is a continuous function, then the value of k is :