Left-hand limit

Definition

The left-hand limit of a function at a particular point is the value that the function approaches as the input approaches that point from the left side. This concept is crucial for understanding the behavior of functions at specific points, particularly when dealing with discontinuities or asymptotes. It allows for the analysis of one-sided limits, which can provide insights into the overall limit of a function when combined with the right-hand limit.

5 Must Know Facts For Your Next Test

1. The notation for the left-hand limit of a function f(x) as x approaches 'a' is written as $$\lim_{x \to a^-} f(x)$$.
2. If the left-hand limit exists and is equal to the right-hand limit, then the overall limit at that point exists.
3. Left-hand limits are particularly useful in analyzing piecewise functions where different rules apply to different intervals.
4. If a function has a jump discontinuity at a point, the left-hand and right-hand limits will be different.
5. In cases where one-sided limits exist but do not match, it indicates that the overall limit does not exist at that point.

Review Questions

• How do you determine if a left-hand limit exists for a given function at a specific point?
  • To determine if a left-hand limit exists for a function at a specific point, evaluate the function as it approaches that point from values less than it. This means calculating $$\lim_{x \to a^-} f(x)$$, where 'a' is the point in question. If this limit results in a finite value, then the left-hand limit exists. If it diverges or doesn't settle on any specific value, then the left-hand limit does not exist.
• Discuss how left-hand limits interact with right-hand limits to determine if an overall limit exists at a certain point.
  • Left-hand limits and right-hand limits are both necessary to determine if an overall limit exists at a specific point. If both limits approach the same value, then we conclude that the overall limit exists and equals that value. However, if they approach different values or one does not exist while the other does, this indicates that no overall limit exists at that point. The agreement or disagreement between these two one-sided limits is crucial for understanding continuity and discontinuities in functions.
• Evaluate the implications of having different left-hand and right-hand limits at a point in relation to continuity.
  • When left-hand and right-hand limits at a certain point yield different values, it signifies a discontinuity in the function at that point. This lack of agreement means that although we can ascertain behavior approaching from both sides, there’s no single output for that specific input value. In practical terms, this can result in scenarios such as jump discontinuities where there are sudden changes in value. Understanding this helps in graphing functions and analyzing their behavior around critical points.

Related terms

right-hand limit:

The right-hand limit of a function at a particular point is the value that the function approaches as the input approaches that point from the right side.

limit: The limit of a function is the value that the function approaches as the input approaches a certain value, regardless of the direction from which it approaches.

discontinuity: A discontinuity is a point at which a function is not continuous, meaning there is a break, jump, or asymptote in the graph of the function.