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

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Answer 1 · 2015-09-21T01:59:46

It is $oo$

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

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

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