Problem 98 - Calculus: Early Transcendentals

Question

Odd function limits Suppose \(g\) is an odd function where \(\lim _{x \rightarrow 1^{-}} g(x)=5\) and \(\lim _{x \rightarrow 1^{+}} g(x)=6 .\) Find \(\lim _{x \rightarrow-1^{-}} g(x)\) and \(\lim _{x \rightarrow-1^{+}} g(x)\).

Step-by-Step Solution

Answer

Question: If the function \(g(x)\) is odd, and we have the following limits as \(x\) approaches 1: \(\lim_{x\to 1^-} g(x) = 5\) and \(\lim_{x\to 1^+} g(x) = 6\), find the limits as \(x\) approaches -1 from the left and right. Answer: Using the properties of odd functions, we found that \(\lim_{x\to -1^-} g(x) = -5\) and \(\lim_{x\to -1^+} g(x) = -6\).

Step 1: Recall the properties of odd functions

An odd function is a function that satisfies the following property: \(g(-x) = -g(x)\) for all values of \(x\) in its domain. This implies that the graph of the function is symmetric with respect to the origin.

Step 2: Use the properties of odd functions to find the limit as x approaches -1 from the left

We are given that \(\lim_{x\to 1^-} g(x) = 5\). Because \(g\) is an odd function, we can use the odd function property to rewrite the limit as \(\lim_{x\to -1^-} g(-x) = -\lim_{x\to -1^-} g(x)\). Now, let's substitute \(-x\) with \(1\): $$\lim_{x\to -1^-} g(-x) = \lim_{x\to 1^-} g(x) = 5$$ Since \(\lim_{x\to -1^-} g(-x) = -\lim_{x\to -1^-} g(x)\), we can then solve for \(\lim_{x\to -1^-} g(x)\) by dividing by -1: $$\lim_{x\to -1^-} g(x) = -\lim_{x\to 1^-} g(x) = -5$$

Step 3: Use the properties of odd functions to find the limit as x approaches -1 from the right

We are given that \(\lim_{x\to 1^+} g(x) = 6\). Similar to what we did in Step 2, we use the odd function property to rewrite the limit as \(\lim_{x\to -1^+} g(-x) = -\lim_{x\to -1^+} g(x)\). Now, we substitute \(-x\) with \(1\): $$\lim_{x\to -1^+} g(-x) = \lim_{x\to 1^+} g(x) = 6$$ Since \(\lim_{x\to -1^+} g(-x) = -\lim_{x\to -1^+} g(x)\), we can solve for \(\lim_{x\to -1^+} g(x)\) by dividing by -1: $$\lim_{x\to -1^+} g(x) = -\lim_{x\to 1^+} g(x) = -6$$

Step 4: Summarize the results

Using the properties of odd functions, we found the following two limits: $$\lim_{x\to -1^-} g(x) = -5$$ and $$\lim_{x\to -1^+} g(x) = -6$$

Key Concepts

Odd Function Properties

Odd functions play a significant role in the realm of mathematics, exhibiting unique characteristics that can simplify complex problems. To identify an odd function, look for symmetry in its graph about the origin, meaning that reflecting the graph across both the x and y axes yields the original function. This critical characteristic is mathematically expressed by the equation \( g(-x) = -g(x) \) for any value of \( x \) within the function’s domain.

As a result of this property, odd functions demonstrate intriguing behavior, such as taking the limit of an odd function at symmetrical points around the origin. Understanding these properties not only demystifies the behavior of odd functions but also provides a powerful tool for solving calculus problems involving limits and symmetry.

Limit of a Function

The limit of a function is a fundamental concept in calculus, capturing the idea of the value that a function approaches as the input approaches a certain point. Whether a function is odd or even, understanding limits is crucial as it unveils the function's behavior near specific points. For instance, the notation \( \lim_{x \to c} g(x) \) signifies the value that \( g(x) \) tends toward as \( x \) approaches \( c \), from either direction unless specified otherwise. This concept is at the heart of numerous calculus operations and further extends into the continuous and discrete realms, integral calculus, and beyond.

Symmetry in Functions

Reflective Symmetry of Functions

Reflective symmetry in functions is an evocative visual concept and a practical analytical tool. With odd functions, symmetry about the origin ensures that for every point \( (x, y) \) on the graph, there is a corresponding point \( (-x, -y) \). This insight simplifies countless calculations, particularly with limits. Such symmetry leads to patterns in behavior as \( x \) approaches specific values, allowing for predictions about the function's behavior at corresponding negative values of \( x \).

Calculus

Calculus is the mathematical study of continuous change, employing concepts like functions, limits, derivatives, and integrals to model and analyze the dynamic world around us. Odd functions and their symmetry properties are particularly important when dealing with integrals over symmetric intervals or finding limits around points of symmetry. The exploration of limits, as illustrated in our example, is one of the introductory steps into the vast and intricate world of calculus, paving the way for concepts such as continuity, rates of change, and the areas under curves.