Limits
Observation of a Graph
Sometimes you will be asked to solve for the limit of a function where the limit does not exist. These mostly will come in the form of a one-sided limit. You can calculate such problems either by substituting in numbers very close to the limit or by using a graph.
Example:
which is undefined and the equation cannot be factored. However, it can be solved by looking at it graphically. At x = 0 the graph diverges into opposite directions. When this happens a limit does not exist. So, 
does not exist.
However, if asked to solve for a one-sided limit, the limit will always exist. The limit will either be from the positive side or the negative side. The positive side is on the right of the graph and the negative side is on the left of the graph.
A limit from the positive side of number “n” is denoted by 
A limit from the negative side of a number “n” is denoted by 
, as can be seen on the graph of 
Other examples:
3.1 
as seen graphically (graph on your TI)
3.2 
as seen graphically
Since the limits do not converge to the same point from the positive or negative sides;
does not exist.
3.3 
as seen graphically
3.4 
as seen graphically
Since the limits converge to the same point from the positive and negative sides;
Greatest integer function: (denoted by [ ])
3.5 
3.6 
3.7 
does not exist.
Try the ones below for practice.
3.8 
as seen graphically
3.9 
as seen graphically
3.10 
as seen graphically
3.11 
