Evaluate the following limit: `lim_(n->oo)(sin(a/(2^n)))/(sin(b/(2^(n\ ))))`

Explanation

`lim_(n->oo)(sin(a/(2^n)))/(sin(b/(2^(n\ ))))`

Take `1/n=h`

Then

`=lim_(h to 0)(sin(a/(2^(1/h))))/(sin(b/(2^((1/h) ))))`

`=(lim_(h to 0) sin(a/(2^(1/h))))/(lim_(h to 0) sin(b/(2^((1/h) ))))`

`=(lim_(h to 0) sin(a/(2^(1/h))). (a)/(2^(1/h)))/(lim_(h to 0) sin(b/(2^((1/h) ))). b/(2^()1/h))`

`=lim_(h to 0)(1xx a/(2^(1/h)))/(1xx b/(2^(1/h)))`

`=a/b`