Limit, Continuity and Differentiability

The value of $\displaystyle\lim_{x \to 0} \frac{\sin 3x}{\tan 5x}$ is

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

Let $f:\mathbb{R}\to\mathbb{R}$ be a twice-differentiable function such that $(\sin x\cos y)\bigl[f(2x+2y)-f(2x-2y)\bigr]=(\cos x\sin y)\bigl[f(2x+2y)+f(2x-2y)\bigr]$ for all $x,y\in\mathbb{R}$. If $f'(0)=\tfrac12$, then the value of $24\,f''\bigl(\tfrac{5\pi}{3}\bigr)$ is:

If the function $f(x)=2x^3 - 9a x^2 + 12a^2 x + 1$, where $a>0$, attains its local maximum and local minimum at $p$ and $q$ respectively, such that $p^2 = q$, then $f(3)$ is equal to:

For $\alpha,\beta,\gamma\in\mathbb{R}$, if $\displaystyle\lim_{x\to0}\frac{x^2\sin(\alpha x)+(\gamma-1)e^{x^2}}{\sin(2x)-\beta x}=3$, then $\beta+\gamma-\alpha$ is equal to:

Let $f(x)=\begin{cases}(1+ax)^{1/x},&x<0,\\1+b,&x=0,\\\displaystyle\frac{\sqrt{x+4}-2}{\sqrt[3]{x+c}-2},&x>0\end{cases}$ be continuous at $x=0$. Then $e^a b c$ is equal to:

Let $a>0$. If the function $f(x)=6x^{3}-45a x^{2}+108a^{2}x+1$ attains its local maximum and minimum values at the points $x_{1}$ and $x_{2}$ respectively such that $x_{1}x_{2}=54$, then $a+x_{1}+x_{2}$ is equal to:

Let $f$ be a differentiable function on $\mathbb{R}$ such that $f(2)=1$ and $f'(2)=4$. If $\displaystyle\lim_{x\to0}\bigl(f(2+x)\bigr)^{3/x}=e^{\alpha}$, then the number of times the curve $y=4x^{3}-4x^{2}-4(\alpha-7)x-\alpha$ meets the $x$-axis is:

If \lim_{x\to1} \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_{n\to\infty} [f(x) - f(x/2^n)] = G(x), then \sum_{r=1}¹⁰ G(r²) is equal to: