Evaluate $\sinh\left(0\right)$ given that the definition of the hyperbolic sine function is, $\sinh\left(x\right)=\frac{e^{x}-e^{-x}}{2}$, and sketch a plausible graph for $y=\sinh(x)$.

Given that a function $h$ satisfies $0\le j\left(x\right)\le 1-\cos x$ for all $x$ near $0$, determine ${\lim }_{x\to 0}j\left(x\right)$.

Evaluate the limit:

$\underset{h\to 0}{\lim }\frac{\tan \left(\tan \left(h\right)\right)}{\tan \left(h\right)}$

Find the limit of the polynomial function $f\left(x\right)=x^3-2x^2+3x-4$ as $x$ approaches $0$.

Determine all vertical asymptotes $x=a$ of the function $f\left(x\right)=\frac{2x+3}{x^4-8x^3+16x^2}$. Evaluate $\underset{x\to a^{+}}{\lim }f\left(x\right)$, $\underset{x\to a_{}^{-}}{\lim }f\left(x\right)$, and $\underset{x\to a}{\lim }f\left(x\right)$ for each value of $a$.

Determine the limit:

$\underset{x\to \infty }{\lim }(\sqrt{x^2+36}-\sqrt{x^2-4})$

Identify the interval of continuity for the function $h\left(x\right)=\frac{4x^2-8x+9}{3x^2+3x+3}$.

Find the limit as $x\to\frac{\pi}{2}$ of the expression $\cos \left(\frac{\pi }{4}\sin \left(\cos x\right)\right)$. Determine if the function is continuous at the point being approached.

Find the value of $a$ that ensures the function $h\left(x\right)=\{\begin{array}{c}\frac{{\sin }^4\left(2x\right)}{x^4},x\ne 0\\ a,x=0\end{array}$ is continuous at $x=0$.