12.2 Techniques for Evaluating Limits

A function is continuous at c if exists and
It can also be written as

Example of Continuous:

Types of Noncontinuous

           


Holes                 






Vertical Asymptotes 







 Piecewise Functions

Techniques for Evaluating Limits

Direct Substitution:

Dividing Out:

Rationalizing:

12.1 Introduction to Limits

 

Definition of Limits- If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, the limit of f(x) as x approaches c is L.

 

 

 

 

 

*Note- the limit doesn't care what is actually happening at c

 

Example-

 

 

Example-

 

 

Even though this graph has a hole at x=1, the graph still approaches 2 from both sides so the limit is still 2. The limit doesn't care what is actually happening at f(1).



 

Example-

 

This limit does not exist because as x approaches 0, f(x) oscilalates between 1 and -1.



Properties of Limits-