One-Sided Limit

Definition

A one-sided limit refers to the value that a function approaches as the input approaches a specific point from one side, either the left or the right. This concept helps to analyze the behavior of functions near points of discontinuity and allows for a more nuanced understanding of limits beyond the standard two-sided limit.

5 Must Know Facts For Your Next Test

1. One-sided limits are denoted as $$\lim_{x \to c^-} f(x)$$ for the left-hand limit and $$\lim_{x \to c^+} f(x)$$ for the right-hand limit.
2. If both one-sided limits exist and are equal, then the two-sided limit exists at that point.
3. One-sided limits are particularly useful for analyzing functions that have points of discontinuity or vertical asymptotes.
4. The concept of one-sided limits extends to infinite limits, where we can consider how functions behave as they approach positive or negative infinity from one side.
5. Understanding one-sided limits helps to clarify situations in piecewise functions, where different expressions are used depending on the input value.

Review Questions

• How do one-sided limits help in understanding functions that exhibit discontinuities?
  • One-sided limits allow us to examine how a function behaves as it approaches a point from either the left or right. This is crucial when dealing with discontinuities because it helps identify whether the function approaches different values from each direction. If the left-hand and right-hand limits do not match at a certain point, it signals a discontinuity in the function at that location.
• Compare and contrast one-sided limits with two-sided limits in terms of their importance in evaluating function behavior.
  • One-sided limits focus on what happens as a function approaches a certain point from one side, while two-sided limits require both sides to approach the same value for the limit to exist. This distinction is essential because even if a two-sided limit does not exist due to discontinuity, one-sided limits can provide valuable information about the behavior of the function around that point. Thus, both concepts are integral to understanding overall limit behavior.
• Evaluate a scenario where both one-sided limits exist but are not equal. What does this imply about the function's continuity and overall limit?
  • If both one-sided limits exist but are not equal, this indicates that there is a discontinuity at that point in the function. Specifically, it shows that while we can determine how the function behaves as it approaches from each side, there is no common value that represents its behavior from both directions. As a result, this means that the overall limit at that point does not exist, further illustrating that continuity is broken.

Related terms

Two-Sided Limit: A two-sided limit is the limit of a function as the input approaches a particular value from both sides, meaning that the limits from the left and right must be equal for the two-sided limit to exist.

Continuity:

A function is continuous at a point if the limit exists at that point and is equal to the function's value at that point, which ties into understanding one-sided limits in analyzing continuity.

Discontinuity:

A discontinuity occurs at a point where a function does not have a limit, which can be identified by examining one-sided limits to see if they differ or do not exist.