Question

How do you find the limit of `(tan^ -1 x)/ (sin^ -1 x+1)` as x approaches 1?

Answer 1

`lim_(x rarr 1) arctan(x)/(arcsin(x) + 1) = pi/(4 + 2pi)`

Explanation:

Since both `arctan(1)` and `arcsin(1)` are defined, `arcsin(1) != -1`, the function is defined and continuous for the range `[1-dx,1+dx]` for a `dx` as small as we want.

Since this function is continuous at that range we can say

`lim_(x rarr 1) f(x)= f(1)`, or in this case

`lim_(x rarr 1) arctan(x)/(arcsin(x) + 1) = (pi/4)/(pi/2 + 1) = (pi/4)/((2+pi)/2) = pi/(4 + 2pi)`