Problem 46 - Calculus Early Transcendentals

Question

A function \(g\) is odd if \(g(-x)=-g(x)\) for all \(x\) in the domain of \(g\). Suppose \(g\) is odd, with \(\lim _{x \rightarrow 2^{+}} g(x)=5\) and \(\lim _{x \rightarrow 2^{-}} g(x)=8 .\) Evaluate the following limits. a. \(\lim _{x \rightarrow-2^{+}} g(x)\) b. \(\lim _{x \rightarrow-2^{-}} g(x)\)

Step-by-Step Solution

Answer

Question: If \(g(x)\) is an odd function, and \(\lim _{x \rightarrow 2^{+}} g(x) = 5\) and \(\lim _{x \rightarrow 2^{-}} g(x) = 8\), find \(\lim _{x \rightarrow -2^{+}} g(x)\) and \(\lim _{x \rightarrow -2^{-}} g(x)\). Answer: The limit \(\lim _{x \rightarrow -2^{+}} g(x) = -5\) and the limit \(\lim _{x \rightarrow -2^{-}} g(x) = -8\).

Step 1: a. Find \(\lim _{x \rightarrow -2^{+}} g(x)\)

Given that \(\lim _{x \rightarrow 2^{+}} g(x) = 5\), we can rewrite this limit as \(\lim _{x \rightarrow -(-2)^{+}} g(-2) = 5\). Using the property of odd functions, \(g(-x) = -g(x)\), we can say that \(g(2) = -g(-2)\). Now, let's substitute \(g(-2)\) in our rewritten limit: \(\lim _{x \rightarrow -2^{+}} g(x) = -\lim _{x \rightarrow 2^{+}} g(x) = -5\). So, the limit \(\lim _{x \rightarrow-2^{+}} g(x) = -5.\)

Step 2: b. Find \(\lim _{x \rightarrow-2^{-}} g(x)\)

Given that \(\lim _{x \rightarrow 2^{-}} g(x) = 8\), we can rewrite this limit as \(\lim _{x \rightarrow -(-2)^{-}} g(-2) = 8\). Using the property of odd functions, \(g(-x) = -g(x)\), we can say that \(g(2) = -g(-2)\). Now, let's substitute \(g(-2)\) in our rewritten limit: \(\lim _{x \rightarrow -2^{-}} g(x) = -\lim _{x \rightarrow 2^{-}} g(x) = -8\). So, the limit \(\lim _{x \rightarrow-2^{-}} g(x) = -8.\)

Key Concepts

Limits

In mathematics, limits help us understand the behavior of functions as they approach a particular input value. In the context of the exercise, we are interested in the limits of an odd function, which is a function defined such that for every value in its domain, the relation \(g(-x) = -g(x)\) holds. Understanding this property allows us to determine the limits by inverse manipulation of inputs.

For odd functions, as you approach a positive value \(p\), the output is the negative of what you would get if approaching \(-p\). Thus, limits for odd functions exploit their symmetry.

We used the given limits \(\lim _{x \rightarrow 2^{+}} g(x)=5\) and \(\lim _{x \rightarrow 2^{-}} g(x)=8\) to find corresponding limits for \(-2\) by understanding that \(g(-2) = -g(2)\). Therefore, it follows that:

• The limit \(\lim _{x \rightarrow -2^{+}} g(x) = -5\)
• The limit \(\lim _{x \rightarrow -2^{-}} g(x) = -8\)

Continuity

Continuity of a function is a property that describes the unbroken nature of a graph. A continuous function can be drawn without lifting your pen from the paper, as each point smoothly connects to the next. When we consider limits, continuity is particularly useful for predicting function behavior near particular points.

However, the presence of different one-sided limits, as shown in the exercise, signals discontinuity at the point \(x=2\) and consequently at \(x=-2\) because the limits as approached from the left and the right are not equal.

• For \(x=2\), the left-hand limit \(\lim_{x \to 2^{-}}g(x)=8\) and the right-hand limit \(\lim_{x \to 2^{+}}g(x)=5\) indicate a jump discontinuity.
• The symmetry of the odd function then means \(x=-2\) also reflects this jump discontinuity with limits:
• \(\lim _{x \rightarrow -2^{+}} g(x) = -5\)
• \(\lim _{x \rightarrow -2^{-}} g(x) = -8\)

Properties of Functions

Understanding the properties of odd functions gives us insight into how they behave across their entire domain. The hallmark trait of odd functions is the symmetric nature around the origin, represented by the requirement \(g(-x) = -g(x)\). This symmetry heavily influences how limits and continuity are analyzed and understood.

When applied to limits, this symmetry means any behavior of the function approaching a positive \(x\) can be mirrored with negative sign as it approaches \(-x\). Using this property allows calculation of unknown limits based on known ones by merely changing signs.

From a continuity perspective, noticing that limits differ as an odd function is approached from opposite sides does not detract from appreciating their overall symmetry. This inverse symmetry is fundamental in analyzing and simplifying problems involving odd functions, offering a unique perspective in understanding their behavior and their solutions.