Problem 32 - 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 \(23-36\). $$\lim _{x \rightarrow-\infty} \frac{\sqrt[3]{x}-5 x+3}{2 x+x^{2 / 3}-4}$$

Step-by-Step Solution

Answer

The limit is \(-\frac{5}{2}\).

Step 1: Identify the Highest Power of x in the Denominator

Examine the denominator \( 2x + x^{2/3} - 4 \). The highest power of \( x \) is \( x^1 \) or simply \( x \).

Step 2: Divide Numerator and Denominator by x

Divide every term in the numerator \( \sqrt[3]{x} - 5x + 3 \) and the denominator \( 2x + x^{2/3} - 4 \) by \( x \). This gives:\[\lim _{x \rightarrow-\infty} \frac{\frac{\sqrt[3]{x}}{x} - \frac{5x}{x} + \frac{3}{x}}{\frac{2x}{x} + \frac{x^{2/3}}{x} - \frac{4}{x}}\]which simplifies to:\[\lim _{x \rightarrow-\infty} \frac{x^{-2/3} - 5 + \frac{3}{x}}{2 + x^{-1/3} - \frac{4}{x}}\]

Step 3: Evaluate Each Term in the Limit as x Approaches -∞

As \( x \rightarrow -\infty \), evaluate the terms containing \( x \):- \( x^{-2/3} \rightarrow 0 \)- \( \frac{3}{x} \rightarrow 0 \)- \( x^{-1/3} \rightarrow 0 \)Thus, the limit becomes:\[\lim _{x \rightarrow-\infty} \frac{0 - 5 + 0}{2 + 0 - 0} = \frac{-5}{2}\]

Step 4: Conclusion

The limit is \(-\frac{5}{2}\), as the non-zero terms dominate the expression when \( x \rightarrow -\infty \).

Key Concepts

Understanding Noninteger Powers

Noninteger powers, such as square roots or cube roots, are powers that are not whole numbers. In this exercise, we encounter the fraction \(x^{2/3}\) and \(\sqrt[3]{x}\), where the powers are noninteger (i.e., two-thirds and one-third, respectively).
Understanding noninteger powers involves recognizing that they represent fractional exponents, such as \(x^{1/3}\), which is equivalent to the cube root of \(x\). These are used to describe varying behaviors of functions, especially as \(x\) approaches extreme values like \(\pm\infty\).

• Noninteger powers often transform into simpler terms once you perform algebraic manipulations like division.
• It's important to manage these powers carefully, as they influence the behavior of the entire expression.

Being comfortable with manipulating expressions with noninteger powers is crucial for understanding more complex phenomena like limits and transformations.

Manipulating Negative Powers

Negative powers, such as \(x^{-2/3}\) or \(\frac{3}{x}\), essentially mean the reciprocal of the positive power. So, \(x^{-a} = \frac{1}{x^a}\). Recognizing and manipulating these can help approach limit problems effectively.
Negative powers often become more prominent when dividing a function by its highest degree term. In our exercise, dividing both the numerator and the denominator by the highest power,\(x^1\), simplifies expressions like \( x^{-2/3} \) while helping identify dominant terms.

• While evaluating limits, it’s common for terms with negative powers to approach zero, simplifying the analysis.
• Ensure precision when handling signs and powers; negative exponents can switch terms from large to small significance with limits.

Understanding negative powers allows us to effectively simplify and evaluate limits, making them essential in analyzing rational functions.

Evaluating Limits of Rational Functions

Evaluating limits of rational functions involves analyzing the behavior of a function as \(x\) approaches a particular value, like \(\pm\infty\). The core technique is to simplify the function, often by dividing the numerator and the denominator by the highest power of \(x\) present in the denominator.
In this exercise, dividing by \(x\) and simplifying enables us to view the expression's contributing terms clearly. When \(x\) heads towards \(-\infty\), terms like \(x^{-2/3}\), \(\frac{3}{x}\), and \(x^{-1/3}\) all approach zero because their denominators grow indefinitely large.

• This helps in isolating dominant terms in the rational function, revealing the limit as some other terms diminish to insignificance.
• The major skill is to manage complex expressions by deciphering which terms dominate and which shrink away.

Mastery in evaluating such limits affords clarity on the behavior of functions over extensive ranges, providing valuable insights into their long-term trends.