What is the limit of (x-cosx/x) as x goes to infinity?

It is $\infty$

Explanation:

We have that

${\lim}_{x \to \infty} \left(x - \cos \frac{x}{x}\right) = {\lim}_{x \to \infty} x - {\lim}_{x \to \infty} \left(\cos \frac{x}{x}\right)$

Hence $- 1 \le \cos x \le 1 \implies - \frac{1}{x} \le \cos \frac{x}{x} \le \frac{1}{x} \implies \left\mid \cos \frac{x}{x} \right\mid \le \frac{1}{x}$

Using the sqeeze theorem we have that

${\lim}_{x \to \infty} \left(\cos \frac{x}{x}\right) \to 0$ so we have that

lim_(x->oo)(x-cosx/x)=lim_(x->oo) x-lim_(x->oo)(cosx/x)= (oo)-0=oo