How to know when to do certain steps to find the limit of an indeterminate form

Question

For the function

f(x) = sqrt(x² - 2) - sqrt(x² - x)

when taking the limit of x as it approaches infinity, if I use a conjugate first, and then divide the numerator and the denominator by the highest power of x, I get 1/2 as x approaches infinity. However, if I divide the numerator and the denominator by the highest power of x first, and then apply the conjugate method, I still get 0/0. Is there any way in determining when to use which method first?

Answer 1

If you end up with indeterminate form like 0/0, you can apply L'hospital rule which means you can take derivative of both numerator and denominator, then you can take the limit of the new expressions.