Evaluating Limits

We’ll now see how to evaluate limits using the following limit properties:

                             

Example 1:     

We’ll do this first using the rules above.   (This will help you see WHY we can take the short-cut we take in redoing this same problem in example 2.)   The red numbers specify the rule used to justify each step.

               

Example 2:   

Notice how tedious the above calculation was.   The above limit properties basically allow us to simply substitute the value 1 into the expression.   Why not just do it in one step?

                                  

Example 3:   

                                           

 Example 4:      

                                               

OOPS!    There seems to be a problem with this one, since the denominator went to 0 (and division by 0 is undefined).   Expressions like 0/0 or infinity/infinity are called indeterminate forms, and mean you’ve hit a dead-end.   We must do some algebra to hopefully find a way around this difficulty…  

If we factor the numerator and denominator, we often can cancel a common factor:
                

After canceling the offending common factor, the denominator no longer goes to 0, and we can substitute the limit value directly into the expression, as before.

It’s instructive to compare the graphs of two functions that are almost identical.  One is the rational function whose limit we just computed:

                                                           

The other is the function we got AFTER we cancelled the common factor of (x-2):

                                                                 

Recall that when taking the limit, the value the function actually attains (or doesn’t!) at the limit point has NO effect on the result of taking the limit.   Look at the above graph, and consider the limit as x->2 geometrically.   When we take the limit as x->2, the result is the natural continuation of the function to that limit point.  Looking at the graph, it’s clear that the natural continuation of the function would “fill in the hole”.