You can use the algebraic technique of factoring to solve “real” limit problems. All algebraic methods involve the same basic idea. When substitution doesn’t work in the original function — usually because of a hole in the function — you can use algebra to manipulate the function until substitution does work (it works because your manipulation plugs up the hole).
Here’s an example of solving a limit by factoring:
Try plugging 5 into x — you should always try substitution first.
Factor:
Cancel the (x – 5) from the numerator and denominator.
Now substitution will work.
= 5 + 5
= 10
And note that the limit as x approaches 5 is 10, which is the height of the hole at (5, 10).