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:

  1. Try plugging 5 into xyou should always try substitution first.

  2. Factor:

  3. Cancel the (x – 5) from the numerator and denominator.

  4. 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).