12.2- Techniques for Evaluating Limits

Limits of Polynomial and Rational Functions

--1. If p is a polynomial function and c is a real number, then  



--2. If r  is a rational function given by r(x)= p(x)/q(x), and c is a real number such that q(c) does not equal 0, then:
     as x approaches c,   lim r(x) = r(c) = p(c)/q(c)

Evaluating Limits by Direct Substitution

      - when evaluating a limit by direct substitution, you insert the c value in for the x value of the function and the resulting number is the limit as x approaches c.

     - This method only works if the function is continuous, ehich means there are no holes in the graph

Evaluating a limit by Dividing Out

  

     - To evaluate a limit by dividing out, you can factor the numerator and divide out the common factors.

     - This particular example has a limit of 7, and has a hole at the point (1,7) because the factor of (x-1) is divided out, and at that point, y=7.

Evaluating a Limit by Rationalizing

    - To evaluate a limit by rationalizing, you have to first rationalize the numerator of the function.

   - In this example, once you get the numerator rationalized, you can plug in the c value for x and that gives the limit as x approaches c, where at c there will be a hole in the graph.

Evaluating One-Sided Limits

To evaluate a one sided limit, you find what y value is approached from either side as x approaches c.  In this graph, as x approaches c (2) from the left, y also approaches 2.  As x approaches c from the right, however, the y value approaches 1.  These can be solved by simply looking at the graph, or applying direct substitution to the 2 functions for their correct sides of the graph.  

Some helpful videos:

http://www.youtube.com/watch?v=0u7NtyGGlv4

http://www.youtube.com/watch?v=mGPyMDi1Mdc

Thanks,

Michael