Question

How do you find the limit of `(arctan(5x))/(sin(7x))` as x approaches 0?

Answer 1

`=5/7`

Explanation:

`lim_(x->0) arctan(5x)/sin(7x) = arctan0/sin0 = 0/0`

This is an indeterminate type so we use the l'Hopitals'Rule. That is, find the limit of the derivative of the top divided by the derivative of the bottom.

`lim_(x->0) (1/(1+25x^2) *5)/(7cos7x)`

`= (1/(1+0) *5)/(7cos0)`

`=(1*5)/(7*1)`

`=5/7`