Problem 26 - 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 \(23-36\). $$\lim _{x \rightarrow \infty} \sqrt{\frac{x^{2}-5 x}{x^{3}+x-2}}$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Simplify the Expression

First, write the given expression: \[\lim _{x \rightarrow \infty} \sqrt{\frac{x^{2}-5 x}{x^{3}+x-2}}.\]Identify the highest power of \(x\) in the denominator, which is \(x^3\). Divide both the numerator and the denominator by \(x^3\):\[\lim _{x \rightarrow \infty} \sqrt{\frac{x^{2}/x^3 - 5x/x^3}{x^3/x^3 + x/x^3 - 2/x^3}}.\]

Step 2: Simplify Each Term

Simplify the terms in the expression:\[= \lim _{x \rightarrow \infty} \sqrt{\frac{x^{-1} - 5x^{-2}}{1 + x^{-2} - 2x^{-3}}}\]This results from dividing each term by \(x^3\).

Step 3: Evaluate Limit of Each Term

Now observe each term in the numerator and denominator as \(x\) approaches infinity:- \(x^{-1}\) approaches \(0\) as \(x \to \infty\).- \(5x^{-2}\) approaches \(0\) as \(x \to \infty\).- \(1\) remains \(1\).- \(x^{-2}\) and \(2x^{-3}\) both approach \(0\) as \(x \to \infty\).Therefore, the limit simplifies to:\[\lim _{x \rightarrow \infty} \sqrt{\frac{0 - 0}{1 + 0 - 0}}\]

Step 4: Calculate the Final Limit

The simplified expression yields:\[= \lim _{x \rightarrow \infty} \sqrt{0}\]Calculate the square root of \(0\):\[= 0\]Thus, the limit of the original expression as \(x\) approaches infinity is \(0\).

Key Concepts

Rational Functions in Calculus

Rational functions are equations represented by the ratio of two polynomials. For example, consider the function \( f(x) = \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials. These functions are significant in calculus since they help us understand various behaviors, especially as \( x \) approaches large values or boundaries.
Rational functions can have different degrees of difficulty depending on the polynomial in the numerator and the denominator. Calculating their limits can sometimes involve more complex techniques due to these variety of polynomial degrees.
When you encounter a rational function problem dealing with limits, a useful strategy is to divide each term by the highest degree of \( x \) found in the denominator. This often simplifies the expression, making it easier to find the limit when \( x \) approaches infinity. In calculus, these techniques help dissect complex problems into simpler parts by focusing on the most significant behavior of each function.

Understanding Infinite Limits

Infinite limits explore what happens to a function's value as the input becomes infinitely large or small. For rational functions, like the one we are working with, finding limits as \( x \) approaches infinity reveals the function’s long-term behavior.
To compute the limit as \( x \to \infty \), first simplify the equation using the highest power of \( x \) from the denominator. For example, if the polynomial in the denominator is \( x^3 \), divide all terms by \( x^3 \). This isolates the terms that "matter" — as in, those that don't vanish at infinity.

• Terms like \( x^{-1} \) or \( 5x^{-2} \) shrink to zero as \( x \to \infty \), because any positive power of \( x \) in the denominator grows indefinitely, wiping out the numerator.
• Anything with zero in the denominator isn't valid, so make sure to check constant terms or terms that don't vanish.

This analysis involves always checking how each term in the function behaves as \( x \to \infty \), leading to a simplified and more manageable final expression to find the limit.

Asymptotic Behavior and Its Importance

An essential aspect of analyzing rational functions is understanding their asymptotic behavior. Asymptotes are lines that a function approaches as the input heads towards infinity or a boundary value, but never actually touches. There are several kinds of asymptotes, including vertical, horizontal, and oblique.
In the given exercise, horizontal asymptotic behavior is of interest. As you investigate \( \lim_{x \to \infty} f(x) \), you see how the function behaves at extreme values of \( x \). If the expression simplifies to a constant, that constant is the horizontal asymptote the function approaches at infinity.

• Horizontal asymptotes tell us about the bounded behavior of functions at infinity.
• In our solved problem, after simplifying, the function approaches zero, indicating \( y=0 \) is a horizontal asymptote.

Understanding this behavior gives insights into the function's characteristics across its domain, ensuring a comprehension of how the function behaves "at the edge." This information is crucial not only for solving limits but also in broader calculus applications like curve sketching and engineering analyses.