12.2 - Techniques for Evaluating Limits

In the last section, we learned how to solve basic limit problems. However, not all limit problems can be solved this easily. To solve these, we must learn other techniques for evaluating limits.

The easiest method of evaluating a limit is to use the technique of direct substitution. Simply substitute the value which x is approaching into the function after the limit and solve (this is the same example from the previous blog post, so you can visit that for a more detailed walkthrough):

Substitute 2 into the equation to find the answer:

3(2) – 2 = 6 – 2 = 4

Although, sometimes direct substitution will not work because it will result in indeterminate form, or zero over zero. When this happens, you must use either the cancellation technique or the rationalizing technique.

The cancellation technique involves factoring the numerator and denominator, canceling out common factors, and performing direct substitution again. Here is Example 3 from page 878 of our textbook, which demonstrates the cancellation technique:

Use the rationalizing technique when the numerator or denominator involve roots. With this method, you multiply both the numerator and denominator by the conjugate and perform direct substitution again. Here is Example 4 from page 879 of our textbook, which demonstrates the rationalizing technique: