Question

How do you evaluate the limit `sin^2(3x)/x^2` as x approaches `0`?

Answer 1

Use `lim_(theta rarr0) sin theta/theta = 1` with `theta = 3x`

Explanation:

`lim_(xrarr0) sin^2(3x)/x^2 = lim_(xrarr0)sin(3x)/x * sin(3x)/x`

`= lim_(xrarr0)sin(3x)/x * lim_(xrarr0) sin(3x)/x`

`= lim_(xrarr0)(3sin(3x))/(3x) * lim_(xrarr0) (3sin(3x))/(3x)`

`= 3lim_(xrarr0)sin(3x)/(3x) * 3 lim_(xrarr0) sin(3x)/(3x)`

`= 3(1)*3(1)`

`= 9`