One-Sided Limits Examples

Let  

Does exist?

A good place to start is to get an idea of what f looks like.

 In order for the limit to exist we need both one-sided limits to exist. As x approaches zero from the left, y gets close to 0 also, therefore . As x approaches zero from the right, y gets close to 1, therefore . Since the left-side limit is 0 and the right-side limit is 1, and 0 and 1 aren't the same, does not exist.

Let  

 Does exist?

We start by graphing f:

In order for the limit to exist we need both one-sided limits to exist and be identical.

As x approaches two from the left, y gets close to 5, therefore

As x approaches two from the right, y still gets close to 5, so

Since the left-hand limit and the right-hand limit both exist and are equal to 5,

= 5.