Determine the indeterminate form and find the limit

Question

Limit as x approaches 0 of arctan(x/5)/arcsin(x^2). 

I used L'Hospital's rules but could not simplify it enough to get an actual answer but the answer that I got was DNE.

Answer 1

Indeterminant 0/0...

H'hopital's rule says:

1/5 sqrt(1-x^2)

--------------------- * -----------------------------

1 + x^2/25 - 2x

now it is not an indeterminant:

1/5 over zero

Answer 2

You got the right answer.

That fraction blows up as x goes to 0.

The derivative of the numerator is 5/(25+x²) which goes to 1/5 as x goes to 0.

The derivative of the denominator is 2x/sqrt(1-x²) which goes to 0 as x goes to 0...therefore, the fraction is unbounded as x goes to 0.