12.1 Limits!

Definition: 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

This is written as

For most cases, you can find the limit by simply substituting c in for x.

Example:

As x approaches 2 on both sides, it is getting closer to 9. This can be confirmed using a graphing utility.


However, substituting the number can give you an undefined answer.
Example:

Since you are finding the limit as x approaches 2,  you can use the table feature on your graphing calculator and create a table that shows the value of the function for x beginning at 1.9 and increasing by 0.01.

From the table, you can estimate the limit to be 5. In this case, you can't obtain the limit by evaluating f(x) when x=2


LIMITS THAT DON'T EXIST

Condition Under Which Limits Do Not Exist

1) f(x) approaches a different number from the right side of c than from the left side of c
2) f(x) increases or decreases without bound as x approaches c
3) f(x) oscillated between two fixes values as x approaches c

In simpler terms:
1) If there is a break in the graph
2) If it goes to infinity
3) If it oscillates (sin)


Example 1: 

In this example, as x approaches 0, y will get closer to two different numbers. Therefore, the limit does not exist.


Example 2:

On the graph 1/x^2, the limit as x approaches 0 does not exist. Although it approaches infinity, infinity is not a real number. It doesn't approach a particular value.


Example 3:

The graph above represents sin(1/x). No matter how close x approaches 0, f(x) always oscillates between 1 and -1. f(x) is not approaching a certain value so the limit does not exist.