Problem 53 - Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Question: Using the property of even functions, find the limits as x approaches -2 from the right (x -> -2^+) and left (x -> -2^-) for the function f(x), given that f(x) is an even function, and the limits as x approaches 2 from the right and left are 5 and 8, respectively. Answer: The limits as x approaches -2 from the right (x -> -2^+) and left (x -> -2^-) for the function f(x) are 5 and 8, respectively.

Step 1: Understand the even function property

Recall that for an even function \(f(-x)=f(x)\). We'll use this property to find the limits as x approaches -2.$

Step 2: Find limit as x approaches -2 from the right (x -> -2^+)

We are given \(\lim _{x \rightarrow 2^{+}} f(x)=5\). We know that the function is even, so \(f(-x)=f(x)\). Now, we can find the limit as \(x\) approaches \(-2\) from the right. \(\lim _{x \rightarrow (-2)^{+}} f(x) = \lim_{-x\to 2^{+}} f(-x) = \lim_{x\to 2^{+}} f(x) = 5\)

Step 3: Find limit as x approaches -2 from the left (x -> -2^-)

We are given \(\lim_{x \rightarrow 2^{-}} f(x)=8\). We know that the function is even, so \(f(-x)=f(x)\). Now, we can find the limit as \(x\) approaches \(-2\) from the left. \(\lim _{x \rightarrow (-2)^{-}} f(x) = \lim_{-x\to 2^{-}} f(-x) = \lim_{x\to 2^{-}} f(x) = 8\)

Step 4: Write down the final answers

a. \(\lim _{x \rightarrow (-2)^+} f(x) = 5\) b. \(\lim _{x \rightarrow (-2)^{-}} f(x) = 8\)

Key Concepts

Limits of Functions

Limits help us understand the behavior of functions as they approach a specific value. The concept of limits is fundamental in calculus and analysis. It allows us to examine how a function behaves near a certain point, even if it doesn't actually reach that point. In mathematical terms, the limit of a function \(f(x)\) as \(x\) approaches \(a\) is written as \(\lim_{x \to a} f(x)\).
In the context of an even function where \(f(-x) = f(x)\), limits can provide critical insights. For example, knowing \(\lim_{x \to 2^{+}} f(x) = 5\) tells us that as \(x\) gets very close to \(2\) from the right side, \(f(x)\) gets very close to \(5\). This behavior is mirrored for \(\lim_{x \to -2^{+}} f(x)\) due to the even nature of the function.
Limits can be one-sided (approaching from the left or right) or two-sided. Understanding both sides can reveal whether a function is continuous at a point and if it aligns smoothly with its graph.

Properties of Functions

Functions have various properties that determine their behavior and characteristics. For even functions, one key property is symmetry about the y-axis. This means the left side of the y-axis mirrors the right side. Mathematically, this is expressed as \(f(-x) = f(x)\) for all \(x\) in their domain.
When dealing with limits and even functions, this symmetrical property greatly simplifies the process. For example, if you know the limit of \(f(x)\) as \(x\) approaches a particular value from one side, you immediately know the corresponding limit from the opposite side without additional calculations.

• Symmetry: Even functions are symmetrical about the y-axis.
• Simplified Calculations: The even property reduces the complexity in evaluating limits.
• Predictability: Knowing one limit often gives you the other for even functions.

Approaching Limits

Approaching limits involves understanding what happens to a function as it gets closer and closer to a specific point. It's like getting nearer to a destination without actually reaching it. With limits, we're interested in 'approaching behavior,' meaning what values the function is getting closer to.
For any function, and specifically for even functions, there can be different scenarios as the function approaches a value:

• Approaching from the Right (\(x \to a^+\)): This means \(x\) comes closer to \(a\) from greater values than \(a\). For example, \(\lim_{x \to 2^{+}} f(x)\) examines how \(f(x)\) behaves as \(x\) approaches \(2\) from the right.
• Approaching from the Left (\(x \to a^-\)): Here, \(x\) approaches \(a\) from smaller values. \(\lim_{x \to 2^{-}} f(x)\) checks behavior from the left.
• Combining Both: If both right and left limits exist and are equal, the two-sided limit exists.

The property you've seen, \(f(-x) = f(x)\), often makes the limits of even functions from opposite directions surprisingly straightforward, ensuring balance and predictability.