14.2.A One-Sided Limits

1. 14.2.A One-Sided Limits Objective: Find one-sided limits.

2. Nonexistence of Limits Revisited… One of the reasons a limit may not exist is if it approaches a different value from both sides. For example, given the function f(x) shown to the right, the limit as x approaches 4 does not exist because the value that the function approaches from the left is not the same as from the right.

3. One-Sided Limits As x approaches 4 from the left, the limit (y-coordinate) appears to approach -1. This is called the left-hand limit and is denoted as… Notice the (-) symbol is behind the 4 and indicates the direction, NOT the sign of the x-value. Even though the limit does not exist, it does have one-sided limits. As x approaches 4 from the right, the limit (y-coordinate) appears to approach 2. This is called the right-hand limit and is denoted as… Once again the (+) symbol behind the 4 indicates direction, not the sign itself.

4. Example #1 As we approach positive 2 from the left, the limit or y-value appears to be -3. Since this is a square root function, the limit can be found by plugging it in:

5. Example #2 The limit as we approach -4 from the right appears to be 0. Once again we can plug in to make sure:

6. Example #3 From the graph below, find the following limits. Note: Even though the limits exist when x approaches 5, f (5) = 2. 4 4 4

7. Example #3 From the graph below, find the following limits. 3 5