To learn more about these concepts go to the website:
 
 

http://www.usd.edu/~jflores/MultiCalc02/WebBook/Chapter_15/Graphics15/Chapter15_2/Html15_2/15.2%20Limits%20and%20Continuity.htm
 
 
 
 

Example:
 
 

 
 

 
 

Note that the denominator becomes zero when we replace x by 3 and y by 4.
 
 

 
 

After canceling the common factor we can find the limit.
 
 

Showing that a given function in 2 variables is discontinuous at a point requires some work.  Because the (x,y) can approach the given point along an infinite number of paths.
 
 

Example:
 
 

 
 

 
 

Then the limit is 1/(1-m). So we get more than one limit as (x,y) approaches (0,0).