How do you show that $\displaystyle\lim_{x\to 0}\frac{\sin(x)}{\sqrt{x\sin(4x)}} $does not exist? How can I show that $\displaystyle\lim_{x\to 0}\frac{\sin(x)}{\sqrt{x\sin(4x)}} $does not exist ?

Let $f(x)=\sin x/\sqrt{x\sin4x}$.  Note that $f$ is an odd function, i.e., $f(-x)=-f(x)$.  Consequently, $\lim_{x\to0}f(x)$, if it exists, must equal $0$.  But if the limit exists, then
$$\left(\lim_{x\to0}f(x)\right)^2=\lim_{x\to0}(f(x))^2=\lim_{x\to0}{\sin^2x\over x\sin4x}={1\over4}\lim_{x\to0}\left(\left(\sin x\over x \right)^2\left(4x\over\sin4x\right)\right)={1\over4}\not=0$$
Thus the limit does not exist.
