Problem 44 - Calculus Early Transcendentals

Question

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 \(\lim _{x \rightarrow a^{+}} f(x)=L\). 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 the limit of a function at a point exists if and only if both the left-hand limit and the right-hand limit at that point exist and are equal. Answer: To prove this statement, we have shown that if the left-hand limit and the right-hand limit both exist and are equal, then the limit at that point exists (Statement a). Conversely, we have also shown that if the limit of a function at a point exists, then both the left-hand limit and the right-hand limit exist and are equal (Statement b). These two statements together establish the fact that the limit of a function at a point exists if and only if both the left-hand limit and the right-hand limit at that point exist and are equal.

Step 1: Show that if left-hand and right-hand limits exist and are equal, then the limit exists

We are given that \(\lim_{x \rightarrow a^{-}} f(x) = L\) and \(\lim_{x \rightarrow a^{+}} f(x) = L\). We need to show that \(\lim_{x \rightarrow a} f(x) = L\). To prove this, let's consider any sequence \((x_n)\) converging to \(a\). Since the left-hand limit exists, for any subsequence \((x_n')\) of \((x_n)\) where \(x_n' < a\), we have \(\lim_{n\rightarrow\infty} f(x_n') = L\). Similarly, since the right-hand limit exists, for any subsequence \((x_n'')\) of \((x_n)\) where \(x_n'' > a\), we have \(\lim_{n\rightarrow\infty} f(x_n'') = L\). Since for any sequence \((x_n)\) converging to \(a\), we can always find subsequences \((x_n')\) and \((x_n'')\) such that they satisfy the above conditions, we have \(\lim_{x \rightarrow a} f(x) = L\). This proves statement a. #Statement b#

Step 2: Show that if the limit exists, then the left-hand and right-hand limits exist and are equal

We are given that \(\lim_{x \rightarrow a} f(x) = L\). We need to show that \(\lim_{x \rightarrow a^{-}} f(x) = L\) and \(\lim_{x \rightarrow a^{+}} f(x)=L\). Let's first prove the left-hand limit: We need to show that for any sequence \((x_n)\) where \(x_n < a\) and converging to \(a\), \(\lim_{n\rightarrow\infty} f(x_n) = L\). Since \(\lim_{x\rightarrow a}f(x)=L\), for the sequence \((x_n)\) converging to \(a\), \(\lim_{n\rightarrow\infty} f(x_n) = L\). So, \(\lim_{x \rightarrow a^{-}} f(x) = L\). Now let's prove the right-hand limit: We need to show that for any sequence \((x_n)\) where \(x_n > a\) and converging to \(a\), \(\lim_{n\rightarrow\infty} f(x_n) = L\). Since \(\lim_{x\rightarrow a}f(x)=L\), for the sequence \((x_n)\) converging to \(a\), \(\lim_{n\rightarrow\infty} f(x_n) = L\). So, \(\lim_{x \rightarrow a^{+}} f(x) = L\). This proves statement b and completes the solution.

Key Concepts

Left-hand limit

The left-hand limit is a fundamental concept in calculus used to determine the behavior of a function as it approaches a certain point from the left side. In mathematical notation, it is expressed as \( \lim_{x \rightarrow a^{-}} f(x) = L \), indicating that as \( x \) approaches \( a \) from values less than \( a \), the function \( f(x) \) gets closer and closer to \( L \).

Understanding the left-hand limit is crucial since it helps in analyzing the continuity and differentiability of functions. If the left-hand limit at a point exists, it means

• There is a consistent, predictable pattern in the function's values when approached from the left side.
• It is possible for a function to have a left-hand limit without an overall limit existing.

One way of verifying the left-hand limit is by taking sequences that approach \( a \) from the left and checking if they lead the function \( f(x) \) consistently towards \( L \). This aligns with the exercise where sequences and subsequences play a role in demonstrating this limit.

Right-hand limit

Just like the left-hand limit, the right-hand limit examines the behavior of a function as it approaches a point, but from the right. It is denoted mathematically by \( \lim_{x \rightarrow a^{+}} f(x) = L \), which means as x approaches \( a \) from values greater than \( a \), the function \( f(x) \) approaches \( L \).

The right-hand limit is essential when evaluating the continuity and overall limits of a function, particularly in pinpointing where potential discontinuities lie. To further understand right-hand limits:

• The function must have predictable behavior when approached from the right.
• Different right and left-hand limits indicate a discontinuity at \( a \).

To prove a right-hand limit, sequences that approach \( a \) from the right are evaluated. As shown in the exercise, combining this with the left-hand limit allows one to confirm the existence of the overall limit of a function at a point.

Sequence convergence

Sequence convergence is an integral part of understanding limits. It refers to whether a sequence approaches a specific value as it progresses infinitely. A sequence \((x_n)\) is said to converge to \(a\) if, as \(n\) becomes extremely large, \(x_n\) gets arbitrarily close to \(a\).

When we relate sequence convergence to limit theory, it allows us to establish the limits of functions by utilizing sequences. In terms of function limits:

• If a sequence of values closer and closer to \( a \) leads \( f(x) \) to \( L \), then the limit at \( x = a \) can be established.
• By examining subsequences (like those approaching from either side of \( a \)), we can validate left-hand and right-hand limits.

In the exercise, sequence convergence helps to demonstrate the existence and equality of left-hand and right-hand limits, therefore establishing the overall limit \( \lim_{x \rightarrow a} f(x) = L \). Knowing how sequences operate provides a more intuitive and practical look at the concept of limits.