Left-Hand Limit

Definition

The left-hand limit of a function at a point is the value that the function approaches as the input variable approaches the point from the left. It represents the behavior of the function as it approaches the point from the left side.

5 Must Know Facts For Your Next Test

1. The left-hand limit is denoted as $\lim_{x\to a^-} f(x)$, where $a$ is the point of interest.
2. The left-hand limit may exist even if the function is not defined at the point of interest.
3. The left-hand limit can be determined numerically by evaluating the function at values approaching the point of interest from the left.
4. The left-hand limit can be determined graphically by observing the behavior of the function as it approaches the point of interest from the left.
5. The existence of the left-hand limit is a necessary condition for a function to be continuous at a point.

Review Questions

• Explain how the left-hand limit is used in the context of finding limits numerically and graphically.
  • In the context of finding limits numerically and graphically (Topic 12.1), the left-hand limit is used to determine the behavior of a function as the input variable approaches a specific point from the left side. Numerically, the left-hand limit can be calculated by evaluating the function at values that are increasingly closer to the point of interest, but from the left. Graphically, the left-hand limit can be observed by examining the behavior of the function's graph as it approaches the point from the left. The left-hand limit provides important information about the function's behavior and is a key component in determining the overall limit of the function at a given point.
• Describe the relationship between the left-hand limit and the concept of continuity (Topic 12.3).
  • The left-hand limit is closely related to the concept of continuity. For a function to be continuous at a point, the left-hand limit and the right-hand limit must both exist and be equal to the function's value at that point. If the left-hand limit exists but is not equal to the function's value at the point, then the function is not continuous at that point. The existence of the left-hand limit is a necessary, but not sufficient, condition for continuity. Understanding the left-hand limit is crucial in determining the continuity of a function at a specific point.
• Analyze how the left-hand limit can be used to determine the behavior of a function near a point of interest, even if the function is not defined at that point.
  • The left-hand limit can provide valuable information about the behavior of a function near a point of interest, even if the function is not defined at that point. By evaluating the left-hand limit, you can determine how the function approaches the point from the left side. This information can be used to make inferences about the function's overall behavior and potential discontinuities. Even if the function is not defined at the point of interest, the left-hand limit can still exist and provide insights into the function's characteristics. Analyzing the left-hand limit is a powerful tool in understanding the function's properties and potential discontinuities.

Related terms

Limit: The value that a function approaches as the input variable approaches a specific point.

Right-Hand Limit: The right-hand limit of a function at a point is the value that the function approaches as the input variable approaches the point from the right.

Continuity: A function is continuous at a point if the left-hand limit and the right-hand limit both exist and are equal to the function's value at that point.