Problem 56 - Calculus: Early Transcendentals

Question

The relationship between one-sided and two-sided limits Prove the following statements to establish the fact that \(\lim f(x)=L\) if and only if \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L . \quad x \rightarrow a^{2}\) a. If \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L,\) then \(\lim _{x \rightarrow a} f(x)=L\) b. If \(\lim _{x \rightarrow a} f(x)=L,\) then \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L\)

Step-by-Step Solution

Answer

Question: Prove that if the left-hand limit and the right-hand limit of a function both exist and are equal, then the two-sided limit also exists and is equal to the one-sided limits. Additionally, if the two-sided limit exists, then both the left-hand limit and the right-hand limit also exist and are equal to the two-sided limit. Answer: This can be proven in two parts. a) If \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L,\) then \(\lim _{x \rightarrow a} f(x)=L\). To show this, select \(\delta = \min\{\delta_{1}, \delta_{2}\}\) such that \(\delta_{1}\) is suitable for the left-hand limit and \(\delta_{2}\) is suitable for the right-hand limit. If \(0 < |x - a| < \delta\), then either \(0 < x - a < \delta\) or \(-\delta < x - a < 0\). In either situation, it implies that \(|f(x) - L| < \epsilon\), meaning the two-sided limit exists and is equal to L. b) If \(\lim _{x \rightarrow a} f(x)=L,\) then \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L\). Using the definition of the limit and considering the cases where \(-\delta < x - a < 0\) and \(0 < x - a < \delta\), it follows that \(|f(x) - L| < \epsilon\), proving that both the left-hand limit and the right-hand limit exist and are equal to L.

Step 1: Definition of the limit

First off, we must recall the definition of a limit: For every positive number \(\epsilon > 0\) there exists a positive number \(\delta > 0\) such that if \(0 < |x - a| < \delta\), then \(|f(x) - L| < \epsilon\).

Step 2: Make use of the left-hand limit and right-hand limit

Since we know that \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L,\) we can apply the definition of a limit individually to the left-hand limit and to the right-hand limit.

Step 3: Choose a suitable \(\delta\) for each limit

For the left-hand limit, for any positive number \(\epsilon > 0\), we can choose a positive number \(\delta_{1} > 0\) such that if \(-\delta_{1} < x - a < 0\), then \(|f(x) - L| < \epsilon\). Similarly, for the right-hand limit, for any positive number \(\epsilon > 0\), we can choose a positive number \(\delta_{2} > 0\) such that if \(0 < x - a < \delta_{2}\), then \(|f(x) - L| < \epsilon\).

Step 4: Chose the minimum of \(\delta_{1}\) and \(\delta_{2}\) as the final \(\delta\)

Now, let \(\delta = \min\{\delta_{1}, \delta_{2}\}\). This means that \(\delta\) is positive because both \(\delta_{1}\) and \(\delta_{2}\) are positive.

Step 5: Show that the chosen \(\delta\) works for the two-sided limit

By choosing this \(\delta\), if we have \(0 < |x - a| < \delta\), then we can say that either \(0 < x - a < \delta\) or \(-\delta < x - a < 0\). In either situation, since \(\delta \leq \delta_{1}\) and \(\delta \leq \delta_{2}\), it implies that \(|f(x) - L| < \epsilon\). Hence, the two-sided limit exists and \(\lim _{x \rightarrow a} f(x)=L\). #b. If \(\lim _{x \rightarrow a} f(x)=L,\) then \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L\)#

Step 6: Definition of the limit

Since we know that \(\lim _{x \rightarrow a} f(x)=L\), we can apply the definition of a limit. For every positive number \(\epsilon > 0\) there exists a positive number \(\delta > 0\) such that if \(0 < |x - a| < \delta\), then \(|f(x) - L| < \epsilon\).

Step 7: Show that the chosen \(\delta\) works for the left-hand limit

Consider the case where \(-\delta < x - a < 0\). Then, \(0 < |x - a| < \delta\), and it follows that \(|f(x) - L| < \epsilon\) by the definition of the limit. Therefore, the left-hand limit exists and \(\lim _{x \rightarrow a^{-}} f(x)=L\).

Step 8: Show that the chosen \(\delta\) works for the right-hand limit

Similarly, consider the case where \(0 < x - a < \delta\). Then, \(0 < |x - a| < \delta\), and it follows that \(|f(x) - L| < \epsilon\) by the definition of the limit. Therefore, the right-hand limit exists and \(\lim _{x \rightarrow a^{+}} f(x)=L\). This concludes the proof of both statements, establishing the relationship between one-sided and two-sided limits.

Key Concepts

One-Sided Limits

Understanding one-sided limits is key to grasping how a function behaves as it approaches a particular point. One-sided limits consider values of a function approaching from just one direction — either from the left or the right.

In mathematical terms:

• A left-hand limit, denoted as \( \lim_{x \to a^-} f(x) \), looks at values of \( f(x) \) as \( x \) approaches \( a \) from the left.
• A right-hand limit, denoted as \( \lim_{x \to a^+} f(x) \), examines values as \( x \) approaches \( a \) from the right.

These one-sided limits help determine the overall behavior of the function at that point. If both the left and right limits exist and are equal, the two-sided limit is said to exist at that point. One-sided limits are particularly useful in piecewise-defined functions or when studying discontinuities.

When you try to establish that both the left-hand and right-hand limits equal \( L \), it indicates consistent behavior as you approach \( a \). This leads us directly into understanding how these combine to form two-sided limits.

Two-Sided Limits

Two-sided limits encompass both one-sided limits and are crucial for a complete picture of a function's behavior at a point. A two-sided limit exists when the function approaches the same value from both the left and right.

If \( \lim_{x \to a^-} f(x) = L \) and \( \lim_{x \to a^+} f(x) = L \), then we can say \( \lim_{x \to a} f(x) = L \). This condition means that as \( x \) approaches \( a \) from any direction, the function \( f(x) \) will converge to the value \( L \).

The existence of a two-sided limit implies continuity at that point, provided that \( f(a) = L \) as well.

• This ensures smooth transitions without jumps or breaks in the graph.
• It provides a unified view, demonstrating that one-sided limits are consistent with each other.

By mastering two-sided limits, you can seamlessly integrate information from left and right and gain insights into the overall behavior of a function.

Epsilon-Delta Definition of a Limit

The epsilon-delta definition provides a rigorous mathematical framework for understanding limits. It requires showing how a function can be made arbitrarily close to a limit by choosing input values sufficiently close to a target value.

This is expressed as:

• For a limit \( L \) to exist at \( a \), and for every \( \epsilon > 0 \) (a desired closeness of \( f(x) \) to \( L \)), there must be a \( \delta > 0 \) (a range around \( a \)) such that whenever \( 0 < |x - a| < \delta \), it follows that \( |f(x) - L| < \epsilon \).

This definition is powerful because:

• It mathematically guarantees the limit.
• It bridges the concepts of proximity in function values and proximity in input values.

Using this method, you can investigate both one-sided and two-sided limits. By selecting appropriate \( \delta \) values, you ensure the correctness of limit claims for any function and make precise the otherwise intuitive notion of 'approaching' a value.