Question

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

Answer 1

`lim_(x rarr 0) (sin^-1 x)/(tan^-1 x)=1`

Explanation:

We evaluate the limit first

`lim_(x rarr 0) (sin^-1 x)/(tan^-1 x)=0/0`

We apply L'Hospitals Rule

`lim_(x rarr 0) (sin^-1 x)/(tan^-1 x)=lim_(x rarr 0) (d/dx(sin^-1 x))/(d/dx(tan^-1 x))=lim_(x rarr 0) ((1/sqrt(1-x^2)))/((1/(1+x^2)))=1`

God bless....I hope the explanation is useful.