Problem 21 - Thomas Calculus

Question

In Exercises \(13-22,\) find the limit of each rational function (a) as \(x \rightarrow \infty\) and \((b)\) as \(x \rightarrow-\infty\) . $$f(x)=\frac{3 x^{7}+5 x^{2}-1}{6 x^{3}-7 x+3}$$

Step-by-Step Solution

Answer

Both limits as \(x \to \infty\) and \(x \to -\infty\) are \(\infty\).

Step 1: Identify Dominant Terms

To determine the limit as \(x \rightarrow \infty\) or \(x \rightarrow -\infty\), we need to identify the dominant terms in the numerator and the denominator. The dominant term is the term with the highest power of \(x\). In the given function \(f(x) = \frac{3x^7 + 5x^2 - 1}{6x^3 - 7x + 3}\), the dominant term in the numerator is \(3x^7\) and in the denominator is \(6x^3\).

Step 2: Approximate Rational Function

Knowing the dominant terms, approximate the function by comparing only the dominant terms. Thus, the function simplifies to \(\frac{3x^7}{6x^3} = \frac{3}{6}x^{7-3} = \frac{1}{2}x^4\).

Step 3: Evaluate Limits

Now evaluate the limits of \(\frac{1}{2}x^4\) as \(x\rightarrow \infty\) and \(x\rightarrow -\infty\).(a) As \(x \rightarrow \infty\), \(x^4 \rightarrow \infty\). Therefore, \(\frac{1}{2}x^4 \rightarrow \infty\).(b) As \(x \rightarrow -\infty\), \(x^4 \rightarrow \infty\) because even powers of negative numbers return positive results. Therefore, \(\frac{1}{2}x^4 \rightarrow \infty\).

Key Concepts

Dominant Terms

When finding limits of rational functions as they approach infinity, identifying the dominant terms is a crucial first step. The dominant term is the term with the highest power of the variable, in this case, \(x\). It dictates the behavior of the function as \(x\) grows larger.For the function \(f(x) = \frac{3x^7 + 5x^2 - 1}{6x^3 - 7x + 3}\), determining dominance is straightforward. In the numerator, the highest power is 7 in the term \(3x^7\). In the denominator, it's 3 in \(6x^3\).

• Why focus on dominant terms? They determine the function's end behavior because for very large or very small values of \(x\), the effect of lower power terms becomes negligible.
• How to identify them: Look at the highest degree of \(x\) in both the numerator and the denominator.

Finding dominant terms helps simplify the function, making further calculations more manageable.

Limits at Infinity

Rational functions often approach a limit as \(x\) tends to infinity or negative infinity. By focusing on the dominant terms, we can more easily determine these limits.For the example function \(f(x) = \frac{3x^7 + 5x^2 - 1}{6x^3 - 7x + 3}\), we approximate it by its dominant terms to form \(\frac{3x^7}{6x^3}\). Simplifying this yields \(\frac{1}{2}x^4\). To then evaluate the integrated limits:

• As \(x \rightarrow \infty\), \(x^4\) becomes infinitely large. Thus, \(\frac{1}{2}x^4\) \( \rightarrow \infty\).
• As \(x \rightarrow -\infty\), \(x^4\) also \(\rightarrow \infty\), reinforcing the result of \(\frac{1}{2}x^4\) \( \rightarrow \infty\).

• Rational functions' limits at infinity give insights into their long-range behavior and potential asymptotes, especially useful for graphing and analysis.

Polynomial Division

Sometimes, evaluating limits at infinity necessitates performing polynomial division. However, in some instances, like with dominant terms, it may not be necessary, though it can offer a more refined view of the function's behavior.For the function \(f(x) = \frac{3x^7 + 5x^2 - 1}{6x^3 - 7x + 3}\), simplifying and evaluating didn't require full polynomial division because simplifying through dominant terms gave the limit at \(x\rightarrow \infty\) effectively.

• Usage: Polynomial division helps find slant asymptotes and precise behavior, typically for error-checking and further analysis.
• Process: Divide the highest degree term of the numerator by the highest degree term of the denominator, proceeding term-by-term as needed.

While in this exercise, dominant terms sufficed, polynomial division remains an essential tool for deeper analysis and understanding of rational functions.