The three types of behavior associated with non existence of limit are

1. When the left hand limit and right hand limit are not equal.

2. Unbounded behavior

3. Oscillating behavior

(1)

Comparing the left hand limit and right hand limit.

Consider an example: .

In this type of function, the two one-sided limits are both finite, yet not equal to each other.

Graph the function.

As tends to  from the left hand side, tends to .

As tends to from the right hand side, tends to .

Since the left hand limit and right hand limit are not equal, limit does not exist.

2. Unbounded behavior

Consider an example: .

This type of function tends to either or .

So this function has a vertical asymptote at .

Graph the function.

The dashed line indicates the vertical asymptote.

As tends to  from the left hand side, tends to .

As tends to from the right hand side, tends to .

Since the left hand limit and right hand limit are not equal, limit does not exist at .

3. Oscillating behavior

This type of function exists when the values of the function appear to be approaching two or more values simultaneously.

Consider an example:  .

Graph the function.

As tends to  from the left hand side, is undefined.

As tends to  from the right hand side, is undefined.

Since the left hand limit and right hand limit is undefined, limit does not exist.

The three types of behavior associated with non-existence of limit are

1. When the left hand limit and right hand limit are not equal.

2. Unbounded behavior.

3. Oscillating behavior