Question

how to find lim x->0 (cos(1/x))/(1+(1/x))

Answer 1

-1=<cos(1/x)<=+1
and lim (1/(1+1/x)=0
then 0*any value =0

Answer 2

change variable
u = 1/x

\[\lim_{u \rightarrow \infty} \frac{\cos u}{1+u}\]
\[-1<\cos u<1\]
thus
\[\lim_{u \rightarrow \infty} \frac{\cos u}{1+u} = \frac{n}{\infty} = 0\]

Answer 3

what does the -1<cosu<1 mean?

Answer 4

@me100mks \(\cos x\) for any real \(x\) always gives a value between \(-1\) and \(1\). it should read \(\forall x\in\mathbb{R},-1\le \cos x\le1\)