Problem 4 - Thomas Calculus

Question

In Exercises \(3-8,\) find the limit of each function (a) as \(x \rightarrow \infty\) and (b) as \(x \rightarrow-\infty .\) (You may wish to visualize your answer with a graphing calculator or computer.) $$f(x)=\pi-\frac{2}{x^{2}}$$

Step-by-Step Solution

Answer

As \( x \to \infty \) and \( x \to -\infty \), the limit of \( f(x) \) is \( \pi \).

Step 1: Identify the Function

The function given is \( f(x) = \pi - \frac{2}{x^2} \). This is a combination of the constant \( \pi \) and the fraction \( \frac{2}{x^2} \), where \( x^2 \) is the square of \( x \).

Step 2: Analyze as \(x \to \infty\)

As \( x \) approaches infinity (\( x \to \infty \)), the term \( \frac{2}{x^2} \) gets smaller because the denominator \( x^2 \) grows very large. Thus, \( \frac{2}{x^2} \to 0 \) as \( x \to \infty \). Therefore, we have \( f(x) = \pi - \frac{2}{x^2} \to \pi - 0 = \pi \) as \( x \to \infty \).

Step 3: Analyze as \(x \to -\infty\)

Similarly, as \( x \) approaches negative infinity (\( x \to -\infty \)), \( x^2 \) is still positive and grows large, making \( \frac{2}{x^2} \) also approach 0. Therefore, \( \frac{2}{x^2} \to 0 \) as \( x \to -\infty \). Thus, \( f(x) = \pi - \frac{2}{x^2} \to \pi - 0 = \pi \) when \( x \to -\infty \).

Step 4: Conclusion

For both \( x \to \infty \) and \( x \to -\infty \), the term \( \frac{2}{x^2} \) approaches 0, causing \( f(x) \) to approach \( \pi \). Therefore, the limit of \( f(x) \) in both directions is \( \pi \).

Key Concepts

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as its input approaches certain points, like infinity in mathematics. When analyzing a function, this behavior helps us understand how the function behaves at the extremes. For the function \[f(x) = \pi - \frac{2}{x^2} \], asymptotic behavior becomes crucial as \(x\) becomes very large or very small. Asymptotes can be horizontal, vertical, or oblique, and they represent lines that the function approaches but never actually reaches.
For instance, as \(x\) approaches both positive and negative infinity in our function, the term \(\frac{2}{x^2}\) diminishes. This causes the function to head towards the horizontal asymptote at \(y = \pi\). Understanding asymptotic behavior is like peeking into the future of a function's behavior, allowing us to predict how it behaves for extreme values, even when it's difficult to graph such large or small values accurately.

Infinite Limits

Infinite limits occur when a function either heads towards infinity, negative infinity, or when it approaches a finite value as the input grows without bound. In this calculus exercise, we focused on finding the limit of the function \(f(x) = \pi - \frac{2}{x^2}\) as \( x \rightarrow \infty \) and \( x \rightarrow -\infty \).

• When \(x\) approaches infinity, \(\frac{2}{x^2}\) tends towards zero, simplifying our function to \(f(x) = \pi\).
• Similarly, as \(x\) approaches negative infinity, \(\frac{2}{x^2}\) still approaches zero because \(x^2\) is always positive, making \(f(x) = \pi\) in this case as well.

Infinite limits help us to understand that even though the values of the function may grow infinitely large or shrink very tiny, the function's output finds a pattern or a number to approach in some manner. This is key for solving calculus exercises involving limits and continuity.

Calculus Exercises

Calculus exercises often challenge students to apply concepts like limits, derivatives, and integrals to solve mathematical problems. These exercises help build a deeper understanding of functions and their behavior under different conditions.
When tackling limits, as in the given exercise, you practice finding how a function behaves when its input reaches extreme values. Points to consider for solving these exercises include:

• Identifying terms that simplify when the variable approaches infinity or an asymptote.
• Recognizing constants in the equation, as they often determine the asymptotic value, like \(\pi\) in our example.
• Using graphing tools can be beneficial for visualizing how the function behaves as it approaches limits.

Solving calculus exercises gives students the opportunity to test their understanding and application of theoretical parts of calculus in practical scenarios.