Problem: Evaluate (if it exists) the limit
Solution: This is a standard problem, we want to find a limit from the right of an expression that exists on a reduced right neighborhood of the limit point. Thus we start by substituting this point into the expression and obtain "zero over zero". For indeterminate ratio we have a standard tool, the l'Hospital rule.
We have again the situation "zero over zero"; however, applying l'Hospital's rule would be suicidal:
We need a better way. One possibility is to separate "bad" (parts that give zero) from "good" and do the nice (non-zero) part separately, which will make l'Hospital feasible:
Now it wasn't that bad, but the roots are still there, since we cannot "kill" them by differentiation. We need an alternative. There is one possibility mentioned in the box "indeterminate ratio", namely cancelling. Here we are lucky and algebra does help.
