What is it in solving the limit (lim)?

Question

Mathematics

Hello. I had to remember the limits the other day. I decided to check my solution in an online solver and saw some trick I didn't know. Reducing x at once in the numerator and denominator. Is it legal at all? What is this approach called? Can always be used?

Answer 1

This technique is called polynomial division. Only now they forgot to remove all the fractions in brackets.

Answer 2

It was they who simply bracketed the maximum power of x separately in the numerator and denominator.