The Limit of a Function
One-Sided Limits
Sometimes, what happens to us at a place depends on the direction we use to approach that place. If we approach Niagara Falls from the upstream side, then we will be 182 feet higher and have different worries than if we approach from the downstream side. Similarly, the values of a function near a point may depend on the direction we use to approach that point. If we let approach 3 from the left ( is close to and , then the values of equal (Fig. 7). If we let approach 3 from the right ( is close to 3 and ), then the values of equal .
On the number line we can approach a point from the left or right, and that leads to one-sided limits.
Definition of Left and Right Limits:
Example 5: Evaluate and .
Solution: The left-limit notation requires that be close to 2 and that be to the left of 2, so .
If is close to 2 and is to the right of 2, then so and .
The graph of is shown in Fig. 8.
If the left and right limits have the same value, , then the value of is close to whenever is close to , and it does not matter if is left or right of so . Similarly, if , then is close to whenever is close to and less than and whenever is close to and greater than , so . We can combine these two statements into a single theorem.
One-sided limits are particularly useful for describing the behavior of functions which have steps or jumps.
To determine the limit of a function involving the greatest integer or absolute value or a multiline definition, definitely consider both the left and right limits.
Practice 3: Use the graph in Fig. 9 to evaluate the one and two-sided limits of at , and .
Find the one and two-sided limits of at and .