Question

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

Answer 1

It is `oo`

Explanation:

We have that

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

Hence `-1<=cosx<=1=>-1/x<=cosx/x<=1/x=>abs(cosx/x)<=1/x`

Using the sqeeze theorem we have that

`lim_(x->oo) (cosx/x)->0` so we have that

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