Chapter 12.2

Techniques for Evaluating Limits

Limits of polynomial and rational functions:

• if p is a polynomial function and c is a real number, then
  
  
  








• If r is a rational function given by r(x)=p(x)/q(x), and c is a real number such that
  then
  
  
  

Direct substitution

To find limits Algebraically (through direct substitution):

1) Factor the numerator and then

2) Cancel out the variables that can cancel

*limits can be found even if a hole does not allow the original problem to work out, but if there's a vertical asymptote, the limit does not exist (DNE)

example:

Continuous and discontinuous functions

For a graph to be continuous, it must fit this fornula:

the left and right side-limits must be equal to each other and must both be equal to the function

(the (+) sign as a superscript means the limit is right-sided and the (-) sign means the limit is left-sided)

Discontinuity is when a function os not continuous

Discontinuity can occur due to three things:

1) holes

2)asymptotes

3) breaks in the graph