Indietro2.4
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Section 2.4: One-Sided Limits
Introduction to One-Sided Limits
In calculus, one-sided limits allow us to analyze the behavior of a function as the input approaches a specific value from only one direction—either from the left or the right. This concept is crucial for understanding discontinuities and the precise definition of limits.
Left-hand limit: The value a function approaches as the input approaches a point from values less than that point.
Right-hand limit: The value a function approaches as the input approaches a point from values greater than that point.
Formal Definitions of One-Sided Limits
Left-Hand Limit: We write and say that the left-hand limit of as approaches is if we can make $f(x)$ arbitrarily close to $L$ by taking $x$ sufficiently close to $c$ with .
Right-Hand Limit: We write and say that the right-hand limit of as approaches is if we can make $f(x)$ arbitrarily close to $L$ by taking $x$ sufficiently close to $c$ with .
Relationship to Two-Sided Limits
The two-sided limit exists if and only if both one-sided limits exist and are equal:
if and only if and
Example: The Function
This function is defined for all .
As (from the left), .
As (from the right), .
Therefore, and .
The two-sided limit does not exist because the left and right limits are not equal.
Evaluating One-Sided Limits from Graphs
One-sided limits can often be determined by analyzing the graph of a function near the point of interest.
To find , observe the values of as approaches from the left.
To find , observe the values of as approaches from the right.
If both one-sided limits exist and are equal, the two-sided limit exists and equals that value.
If the one-sided limits are not equal, the two-sided limit does not exist.
Example Table: Evaluating Limits and Function Values from a Graph
Expression | Interpretation | How to Find |
|---|---|---|
Left-hand limit at | Approach from values less than 2 | |
Right-hand limit at | Approach from values greater than 2 | |
Two-sided limit at | Exists only if both one-sided limits are equal | |
Function value at | Read directly from the graph |
Piecewise Functions and One-Sided Limits
Piecewise functions often have different definitions on either side of a point, making one-sided limits especially important.
For , to find , use the formula; for , use the formula.
If the left and right limits are not equal, does not exist.
Limits Involving Absolute Value and Discontinuity
Functions involving absolute value or piecewise definitions can have different left and right limits at a point, leading to discontinuities.
For , as , the left and right limits may differ due to the absolute value in the denominator.
If , then does not exist.
Summary Table: Existence of Limits
Condition | Does Exist? |
|---|---|
Both and exist and are equal | Yes |
One or both one-sided limits do not exist, or they are not equal | No |
Key Takeaways
One-sided limits are essential for analyzing discontinuities and defining the existence of a two-sided limit.
Always check both one-sided limits when the function is not continuous or is defined piecewise.
Graphical analysis is a powerful tool for evaluating one-sided limits.
