what is the value of the limit as the function cosx/x tends to infinity??...
The squeeze theorem is necessary.
Recall that $\displaystyle -1\leqslant\cos x\leqslant 1$.
Modifiying the inequality, we see that $\displaystyle -\frac{1}{x}\leqslant\frac{\cos x}{x}\leqslant\frac{1}{x}$
Taking the limit of each of these terms as x approaches infinity, we see that
$\displaystyle \lim_{x\to\infty}-\frac{1}{x}\leqslant\lim_{x\to\infty}\frac{\cos x}{x}\leqslant\lim_{x\to\infty}\frac{1}{x}\implies 0\leqslant\lim_{x\to\infty}\frac{\cos x}{x}\leqslant0$.
This then suggests that $\displaystyle \color{red}\boxed{\lim_{x\to\infty}\frac{\cos x}{x}=0}$
Does this make sense?