Problem: Evaluate (if it exists) the limit
Solution: This is a standard problem, we want to find a limit of an expression that exists on a neighborhood of the limit point. Thus we start by substituting infinity into the expression.
This is an indeterminate product. The recommended method from that box is to change the product into a ratio. Which part do we "put under"? The better candidate is x, since after putting under it becomes x−1, which is not markedly worse and it is easy to differentiate. Why are we talking about differentiating? We expect an indeterminate ratio and with the functions we have here, the l'Hospital approach looks best.
Things went as expected and the l'Hospital rule did improve the problem a lot. Now we have an "infinity over infinity" situation, which could be again solved using the l'Hospital's rule, but here even easier trick is to cancel x2 in the expression, since we want the limit at infinity of a ratio of polynomials.
