Problem 25 - 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 \(23-36\). $$\lim _{x \rightarrow-\infty}\left(\frac{1-x^{3}}{x^{2}+7 x}\right)^{5}$$

Step-by-Step Solution

Answer

The limit is \(+\infty\).

Step 1: Identify the Highest Power in the Denominator

First, identify the highest power of \(x\) in the denominator of the function \(f(x) = \left(\frac{1-x^{3}}{x^{2}+7x}\right)^{5}\). The highest power in the denominator \(x^{2} + 7x\) is \(x^{2}\).

Step 2: Divide Numerator and Denominator by \(x^2\)

Next, divide both the numerator and the denominator by the highest power identified in Step 1, which is \(x^2\): \[\frac{1-x^{3}}{x^{2}+7x} = \frac{\frac{1}{x^2} - \frac{x^3}{x^2}}{1 + \frac{7x}{x^2}} = \frac{\frac{1}{x^2} - x}{1 + \frac{7}{x}}.\] Simplify this expression as:\[\frac{\frac{1}{x^2} - x}{1 + \frac{7}{x}}.\]

Step 3: Evaluate the Limit as \(x\) Approaches \(-\infty\)

Evaluate the limit of the simplified expression as \(x\) approaches \(-\infty\):\[\lim_{x \to -\infty} \frac{\frac{1}{x^2} - x}{1 + \frac{7}{x}}.\] Since \(\frac{1}{x^2}\) and \(\frac{7}{x}\) approach 0 as \(x\) approaches \(-\infty\), the expression simplifies to:\[\lim_{x \to -\infty} \frac{-x}{1} = \lim_{x \to -\infty} -x.\] As \(x\) approaches \(-\infty\), \(-x\) approaches \(+\infty\). Thus, the limit of the expression \(\frac{1-x^3}{x^2+7x}\) is \(+\infty\).

Step 4: Evaluate the Function with the Limit

Now raise the result from Step 3 to the power of 5, since the entire expression is raised to the power of 5: \[\lim_{x \to -\infty} \left( \frac{1-x^{3}}{x^{2} + 7x} \right)^{5} = \lim_{x \to -\infty} (+\infty)^{5} = +\infty.\] Therefore, the limit of the original function as \(x\) approaches \(-\infty\) is \(+\infty\).

Key Concepts

Rational Functions

A rational function is any function that can be expressed as the ratio of two polynomials. This means it has the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). These functions are similar to ratios we work with in numbers, but are more complex because they involve variables raised to various powers.
Understanding rational functions is crucial because they serve as the foundation for addressing more complex mathematical functions and problems. When evaluating the limits of such functions, especially as \( x \) approaches infinity or negative infinity, we look at the behavior of the highest degree term, as it tends to dominate the function's behavior.

Noninteger Powers

Noninteger powers refer to exponents that are fractions or decimals, such as \( x^{3/2} \) or \( x^{0.5} \). These powers may represent roots or other transformations of numbers that aren't as straightforward as integer powers. Noninteger powers are significant in calculus because they often appear in limits and derivatives.
Solving equations with noninteger powers requires identifying the impacts of these fractional exponents within an expression. Since they can drastically change the shape and direction of graphs, understanding how noninteger powers behave in limits provides deeper insight into the overall behavior of functions.

Negative Powers

Negative powers, such as \( x^{-1} \) or \( x^{-3} \), indicate reciprocal relationships. A negative exponent means \( x^{-n} \) is equivalent to \( \frac{1}{x^n} \). These are very useful when working with limits and simplifying expressions within mathematical functions.
In calculus, negative powers can simplify complex fractions, making the evaluation of limits more manageable. As \( x \) approaches infinity or negative infinity, terms with negative exponent powers \( \frac{1}{x^n} \) tend toward zero, simplifying the expression significantly and often revealing the true behavior of the function.

Infinity Limits

Infinity limits examine the behavior of a function as the variable \( x \) approaches very large positive or very large negative values, often represented as \( x \to \infty \) or \( x \to -\infty \). Understanding these limits is critical because they tell us how a function behaves at its extremes, regardless of finite points. This can show trends or asymptotic behavior that might not be apparent otherwise.
In the given exercise, analyzing the function \( \left(\frac{1-x^{3}}{x^{2}+7x}\right)^{5} \) as \( x \to -\infty \) involved recognizing the dominant behavior due to higher powers of \( x \) and simplifying the problem to evaluate the limit. This approach leads to determining the overall behavior as the function climbs towards infinity. It highlights the importance of infinity limits for understanding the comprehensive range of rational functions.