Simplify the expression with infinity $sum\left(n=\frac{0^{\infty }\cdot 4}{n\left(n+1\right)}\right)$

Evaluate the limit

$\lim_{x\to0}\left(\frac{\sin\left(3x\right)}{\tan\left(4x\right)}\right)$

Evaluate the limit

$\lim_{x\to0}\left(\frac{x^2}{1-\cos\left(x\right)}\right)$

Evaluate the limit

$\lim_{x\to0}\left(\frac{\sqrt{5+x}-\sqrt{5}}{x}\right)$

Evaluate the limit

$\lim_{x\to10}\left(\frac{\sqrt{x+6}-4}{x-10}\right)$

Evaluate the limit

$\lim_{z\to10}\left(\frac{\sqrt{z+6}-4}{z-10}\right)$

Evaluate the limit

$\lim_{x\to0}\left(\frac{5^x-1}{\ln\left(1+x\right)}\right)$

Evaluate the limit

$\lim_{x\to0}\left(\frac{5x}{\ln\left(1-3x\right)}\right)$