12.2: Evaluating Limits

There are three main ways to evaluate limits.

1) direct substitution (plug-in) 

2) cancellation technique (factor and cancel) 

3) rationalizing technique (multiply by the conjugate-only used with problems with radicals) 

You should use the direct substitution first. If using direct substitution gives you 0/0, then it is in intermediate form. If it is in intermediate form, then you have to either use the cancellation technique or the rationalizing technique to get a new equation. Then plug the number from the original equation into the new equation. 

Example of the cancellation technique

 

Example of the rationalizing technique

One-sided limits

Sometimes there are problems like this:

In these types of problems, you have to plug it in to both eqautions. If it the result is the same for each equation then there is a limit. In this case, the limit is 3. If the result for each eqaution is different, then it is a one sided equation and you can solve for each side separately. 

In other cases, the problem has a f(x) in it and you have to first solve the function before solving the main equation to find the limit. 

Math joke of the day: