Question

Question #d78f0

Answer 1

`=0`

Explanation:

`lim_(x to oo) (sin^2x)/(x^2+1)`

We don't need to do very much here. `sin x` is a periodic and continuous function such that `sin (x) in [-1,1] , x in (-oo, oo)`

we can lift it out of the limit so that

`lim_(x to oo) (sin^2x)/(x^2+1)`

`= (lim_(x to oo) sin^2x)/(lim_(x to oo)(x^2+1))`

`= lim_(x to oo) sin^2x lim_(x to oo) 1/(x^2+1)`

and the limit of the quotient is the quotient of the limits, where the limits are known, so...
`= lim_(x to oo) (sin^2x)(lim_(x to oo) 1)/(lim_(x to oo) ( x^2+1))`

`= (sin^2x) ( 1)/(oo) = 0`