Problem 30 - Thomas Calculus

Question

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of \(x :\) Divide numerator and denominator by the highest power of \(x\) in the denominator and proceed from there. Find the limits in Exercises \(23-36\). $$\lim _{x \rightarrow \infty} \frac{x^{-1}+x^{-4}}{x^{-2}-x^{-3}}$$

Step-by-Step Solution

Answer

The limit is infinity, \(\infty\).

Step 1: Identify the Highest Power in the Denominator

First, take a look at the powers of \(x\) in the denominator of the function \(\frac{x^{-1}+x^{-4}}{x^{-2}-x^{-3}}\). The highest power in the denominator is \(x^{-2}\).

Step 2: Divide Numerator and Denominator by the Highest Power

Divide every term in both the numerator and the denominator by \(x^{-2}\), the highest power in the denominator. This gives us: \[\frac{\frac{x^{-1}}{x^{-2}} + \frac{x^{-4}}{x^{-2}}}{\frac{x^{-2}}{x^{-2}} - \frac{x^{-3}}{x^{-2}}} = \frac{x^{1} + x^{-2}}{1 - x^{-1}}.\]

Step 3: Simplify the Expression

After performing the division, simplify the expression: \[\frac{x + x^{-2}}{1 - x^{-1}}.\]

Step 4: Evaluate Each Term as x Approaches Infinity

As \(x\) approaches infinity, examine the behavior of each term: - \(x \rightarrow \infty\)- \(x^{-2} \rightarrow 0\) and- \(x^{-1} \rightarrow 0\)Thus, the expression simplifies to \(\frac{\infty + 0}{1 - 0} = \infty\).

Step 5: Conclusion

Since the resulting expression is \(\infty\), the limit is marginalized by the behavior of \(x\) in the numerator, leading to the conclusion that the limit is infinity.

Key Concepts

Rational Functions

Rational functions are a key concept in algebra and calculus. They are expressions of the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. These functions can often be complex, featuring various powers of \( x \) in both the numerator and the denominator. Why are they important? Well, rational functions appear frequently when modeling real-world systems, like response times of circuits or fluid dynamics.

To analyze rational functions, especially when evaluating limits as \( x \) approaches infinity, it's helpful to focus on the highest power of \( x \) in the denominator. This step simplifies comparative growth between the numerator and denominator.

The exercise asks to determine the limit of a rational function as \( x \) approaches infinity. By dividing each term by the highest power in the denominator, we simplify the form to make it more approachable. This method highlights how the behavior of higher powers impacts the limit outcome.

Noninteger Powers

Noninteger powers involve expressions where \( x \) is raised to a non-whole number, like fractions or decimals. They broaden the scope of mathematical functions by allowing more flexibility in modeling real-world scenarios. Noninteger powers make polynomials more versatile and are used in many scientific and engineering contexts.

In the context of limits, noninteger powers behave differently as \( x \) approaches infinity. By focusing on noninteger powers, students can learn to predict how these terms behave compared to integer powers. They often diminish or grow at rates between integer powers, which is crucial in calculating limits and evaluating the behavior of functions as variables increase.

Negative Powers of x

Negative powers of \( x \) signify forming a reciprocal relationship with \( x \). Essentially, a term like \( x^{-n} \) can be rewritten as \( \frac{1}{x^n} \). Understanding negative powers is essential as these terms often shrink their value as \( x \) becomes larger. In the limit calculation, terms with negative powers tend towards zero.

In the original exercise, terms like \( x^{-1} \) and \( x^{-4} \) are crucial. As \( x \) goes to infinity, these terms approach zero, simplifying the rational function considerably. Recognizing this helps students understand how despite the presence of seemingly complex expressions, the simplification process makes evaluating limits straightforward.

• Positive powers grow as \( x \) increases.
• Negative powers decrease, helping simplify limits.

Negative powers are a powerful tool when determining dominant terms in large expressions.