Problem 43 Once you know \(\lim _{x \righta... [FREE SOLUTION]

Chapter 2: Problem 43

Once you know \(\lim _{x \rightarrow a^{+}} f(x)\) and \(\lim _{x \rightarrow a} f(x)\) at an interior point of the domain of \(f,\) do you then know \(\lim _{x \rightarrow a} f(x) ?\) Give reasons for your answer.

Short Answer

Expert verified

No, you need both directional limits (right and left) to determine the two-sided limit.

Step by step solution

Understand the Definitions

We need to understand what \( \lim_{x \to a^{+}} f(x) \) and \( \lim_{x \to a} f(x) \) mean. \( \lim_{x \to a^{+}} f(x) \) represents the limit of \( f(x) \) as \( x \) approaches \( a \) from the right side, while \( \lim_{x \to a} f(x) \) represents the two-sided limit, which means \( x \) approaches \( a \) from both sides, left and right.

Analyze given Information

Given \( \lim_{x \to a^{+}} f(x) \), we know how the function behaves as \( x \) approaches \( a \) from the right. Also, given \( \lim_{x \to a} f(x) \), we are trying to find the overall limiting behavior at \( x = a \) from both sides.

Relationship Between Right-hand, Left-hand, and Two-sided Limits

For \( \lim_{x \to a} f(x) \) to exist, the right-hand limit \( \lim_{x \to a^{+}} f(x) \) and the left-hand limit \( \lim_{x \to a^{-}} f(x) \) must both exist and be equal. However, if we only know \( \lim_{x \to a^{+}} f(x) \), we cannot deduce \( \lim_{x \to a^{-}} f(x) \). Therefore, without additional information about \( \lim_{x \to a^{-}} f(x) \), we cannot conclude the existence of \( \lim_{x \to a} f(x) \).

Conclusion

Knowing only \( \lim_{x \to a^{+}} f(x) \) and \( \lim_{x \to a} f(x) \) does not provide enough information about \( \lim_{x \to a} f(x) \) as it lacks the left-side limit. The existence of \( \lim_{x \to a} f(x) \) requires both directional limits to be equal, and the left-hand limit is unknown.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Right-hand Limit

A right-hand limit focuses on how a function behaves as the input approaches a specific point from the right side. It is represented as \( \lim_{x \to a^{+}} f(x) \). Here, the small positive sign indicates that we are only interested in values of \( x \) that are slightly greater than \( a \). This concept is useful in examining behaviors of functions particularly when the function changes abruptly or has discontinuities at that point.
To find the right-hand limit, you imagine that you are approaching the point \( a \) from values that are slightly larger than \( a \). For instance, this is akin to walking along a path and approaching a sign from the right.

Left-hand Limit

The left-hand limit involves approaching a particular point from values that are less than the point of interest. It is denoted as \( \lim_{x \to a^{-}} f(x) \), with the negative sign showing we consider values smaller than \( a \). This highlights how the function behaves when coming from the left side.
Understanding the left-hand limit is crucial when examining scenarios where you might approach a discontinuity or abrupt change from the left. Think of it as observing how a person moves toward a boundary from the left direction.

A left-hand limit exists if, as \( x \) gets infinitely close to \( a \) from the left, the function approaches a specific value.
This value may differ from either the right-hand limit or any actual function value at \( a \).

Two-sided Limit

A two-sided limit combines the observations from both the right-hand and the left-hand limits to determine the overall behavior of a function as \( x \) approaches a particular point. It is symbolized as \( \lim_{x \to a} f(x) \), without any direction specified.
For a two-sided limit to exist, both the right-hand and left-hand limits must independently exist and be equal at the point \( a \). This is a key condition for the existence of a two-sided limit.

If either of the one-sided limits does not exist or if they are not equal, the two-sided limit does not exist.
This concept is particularly important when dealing with continuous functions, as continuity at a point essentially means the function values agree from both directions.

Understanding two-sided limits allows students to grasp when a function appears smooth or continuous at a specific point and when more complex changes in behavior may occur.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Once you know \(\lim _{x \rightarrow a^{+}} f(x)\) and \(\lim _{x \rightarrow a} f(x)\) at an interior point of the domain of \(f,\) do you then know \(\lim _{x \rightarrow a} f(x) ?\) Give reasons for your answer.

Short Answer

Step by step solution

Understand the Definitions

Analyze given Information

Relationship Between Right-hand, Left-hand, and Two-sided Limits

Conclusion

Key Concepts

Right-hand Limit

Left-hand Limit

Two-sided Limit

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Probability and Statistics

Statistics

Decision Maths

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.