Question

Prove: If \(-\infty

Short answer

Question: Prove that the overall limit of a function exists in the extended reals if and only if the left limit and the right limit both exist in the extended reals and are equal. Answer: To prove this, we need to show that (1) if the left and right limits exist in the extended reals and are equal, then the overall limit exists in the extended reals; and (2) if the overall limit exists in the extended reals, then the left and right limits must also exist and be equal in the extended reals. By showing both parts, we can conclude that the overall limit exists if and only if the left and right limits exist and are equal in the extended reals.

Step-by-step solution

Proving the limit exists when both(left and right) limits exist and are equal

Assume that \(\lim _{x \rightarrow x_{0}^-} f(x) = L_1\) and \(\lim _{x \rightarrow x_{0}^+} f(x) = L_2\) both exist in the extended reals and are equal, i.e., \(L_1 = L_2\). Here, we aim to prove that \(\lim _{x \rightarrow x_{0}} f(x)\) also exists in the extended reals and is equal to \(L_1\) and \(L_2\). Let \(\epsilon > 0\) be arbitrary. Since both the left and right limits exist and are equal, there must be a \(\delta_1 > 0\) such that for any \(x\) satisfying \(0 < |x - x_{0}| < \delta_1\), we have: 1) If \(x < x_{0}\), then \(|f(x) - L_1| < \epsilon\) 2) If \(x > x_{0}\), then \(|f(x) - L_2| < \epsilon\) Let \(\delta = \min\{\delta_1\}\). Now, if \(0 < |x - x_{0}| < \delta\), we have: 1) If \(x < x_{0}\), then \(|f(x) - L_1| < \epsilon\) 2) If \(x > x_{0}\), then \(|f(x) - L_2| < \epsilon\) Since \(L_1 = L_2\), we can conclude that \(\lim_{x \rightarrow x_{0}} f(x) = L_1 = L_2\) exists in the extended reals.

Proving both(left and right) limits exist and are equal when the overall limit exists

Assume that \(\lim _{x\to x_0}f(x)=L\) exists in the extended reals. We aim to prove that both the left and right limits exist in the extended reals and are equal to the overall limit. Let \(\epsilon > 0\) be arbitrary. Since the overall limit exists, there must be a \(\delta > 0\) such that for any \(x\) satisfying \(0 < |x - x_{0}| < \delta\), we have \(|f(x) - L| < \epsilon\). Now, consider any \(x\) satisfying \(0 < |x - x_{0}| < \delta\): 1) If \(x < x_{0}\), then \(|f(x) - L| < \epsilon\). Thus, \(\lim_{x\to x_0^-}f(x)=L\) exists in the extended reals. 2) If \(x > x_{0}\), then \(|f(x) - L| < \epsilon\). Thus, \(\lim_{x\to x_0^+}f(x)=L\) exists in the extended reals. Since both the left and right limits exist and are equal to the overall limit \(L\), we have proven the second part of the statement. Thus, we can conclude that \(\lim _{x \rightarrow x_{0}} f(x)\) exists in the extended reals if and only if \(\lim _{x \rightarrow x_{0}^-} f(x)\) and \(\lim _{x \rightarrow x_{0}^+} f(x)\) both exist in the extended reals and are equal, in which case, all three are equal.

Key concepts

Understanding Extended Reals

When we talk about limits, sometimes they approach boundaries like positive or negative infinity. The set of extended real numbers incorporates these infinities into the real number system. It includes all real numbers plus "+∞" and "−∞."
This helps in analyzing behaviors of functions that grow indefinitely or decrease without bound.

• Enabling calculus concepts to embrace unbounded behavior.
• Helping us evaluate limits approaching infinity.
• A vital component in understanding discontinuities.

By using extended reals, we can discuss limits of functions that might not settle into a real number but approach these infinite concepts.

Exploring Left-Hand Limit

The left-hand limit of a function is focused on the behavior of the function as it approaches a certain point from the left side. Mathematically, it's written as \(\lim_{x \to x_0^-} f(x)\).
Here's what you need to know:

• Only considers values of \(x\) less than \(x_0\).
• Focuses on approaching the point from the negative side.
• A crucial check when determining overall limit existence.

If the left-hand limit exists within the extended reals, it might be a finite number or even \(-∞\) or \(+∞\), depending on the function’s nature.

Understanding Right-Hand Limit

The right-hand limit looks at how a function behaves as it nears a specific point from the right side, expressed as \(\lim_{x \to x_0^+} f(x)\).
Some key things to remember:

• Examines values of \(x\) greater than \(x_0\).
• Approaches from the positive direction.
• Helps confirm overall limit existence.

In extended reals, the right-hand limit can be a real number, \(+∞\), or \(-∞\). Both the left-hand and right-hand limits need to be equal for the overall limit to exist.

Epsilon-Delta Definition in Limits

The epsilon-delta definition solidifies our understanding of limits by creating a rigorous mathematical framework. It's all about ensuring the function values get arbitrarily close to the limit. Here's how it works:

• \(\epsilon\) represents an arbitrary small positive number, describing closeness.
• \(\delta\) is a small positive number ensuring \(f(x)\) stays close to \(L\) when \(x\) is near \(x_0\).

If for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that whenever \(0 < |x - x_0| < \delta\), \(|f(x) - L| < \epsilon\), the limit is confirmed. This concept is foundational in proving limits and ensures precise control over function behavior near a point.