Question

Given `(sin x)^(3x^2)` how do you find the limit as x approaches 0?

Answer 1

`1`

Explanation:

`(sin x)^(3x^2) = (1+sinx - 1)^(3x^2) =`
`=1+3x^2(sinx-1)+3x^2(3x^2-1)(sinx-1)^2/(2!)+cdots=`
`1+x^2f(x)` so

`lim_(x->0)(sin x)^(3x^2) = 1+lim_(x->0)x^2 cdot lim_(x->0)f(x) = 1 +0 cdot 0 = 1`