lim as x approaches 0+ of (1/square root of x) - (1/square root of x^2 + x)

Question

I can punch this in to my calculator and get f(x) is equal to zero. But I need to see the steps without using l'hospitals role.

Answer 1

Let x=h² where |h| is very small. Then we have 1/h-1/√(h⁴+h²)=1/h-1/√(h²(1+h²)).

This is 1/h-(1/h)(1+h²)⁻⁰⋅⁵=1/h-(1/h)(1-½h²)=1/h-1/h+½h=½h to first order approximation. So as h→0, x→0⁺ and the limit →0.