Proceeding from Cauchy's definition of the limit of a function prove that :
$\underset{x\to +\infty }{\lim }{a}^x=+\infty \left(a>1\right)$;

Find the limits:

$\underset{x\to 0}{\lim }\frac{\cos x+4\tan x}{2-x-2x^4}$

Find the limits:

$\underset{x\to -2}{\lim }\frac{2x^2+5x-7}{3x^2-x-2}$

Find the limits:

$\underset{x\to 1}{\lim }\frac{\sqrt{5-x}-2}{\sqrt{2-x}-1}$

Find the limits:

$\underset{x\to \infty }{\lim }\frac{2x^2-5x+4}{5x^2-2x-3}$;

Find the limits:

$\underset{x\to \infty }{\lim }\left(\sqrt{x^2+1}-\sqrt{x^2-1}\right)$

Find the limits:

$\underset{x\to 0}{\lim } \left(\frac{1-2x}{\sqrt[3]{1+8x^3}}+2^{-x}\right)$

Find the limits:

$\underset{x\to 0}{\lim }\frac{\sqrt{x+4}-2}{\sin 5x}$