I A Maron Solutions for Chapter: Introduction to Mathematical Analysis, Exercise 9: The Limit of a Function

Prove that $\underset{x\to \infty }{\lim }\cos x$ does not exist.

Using the sequences of the roots of the equations $\sin \left(\frac{1}{x}\right)=1$ and $\sin \left(\frac{1}{x}\right)=-1$, show that the function $f\left(x\right)=\sin \left(\frac{1}{x}\right)$ has no limit as $x\to 0.$

Proceeding from Cauchy's definition of the limit of a function prove that :
$\underset{x\to 1}{\lim }\frac{x-1}{\sqrt{x}-1}=2$;

Proceeding from Cauchy's definition of the limit of a function prove that :
$\underset{x\to 0}{\lim }\sin x=0$;

Proceeding from Cauchy's definition of the limit of a function prove that :
$\underset{x\to 0}{\lim }\cos x=1$;

Proceeding from Cauchy's definition of the limit of a function prove that :
$\underset{x\to +\infty }{\lim }\frac{2x-1}{3x+2}=\frac{2}{3}$;

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

Proceeding from Cauchy's definition of the limit of a function prove that :
$\underset{x\to \infty }{\lim }\frac{\sin x}{x}=0$ ;