Problem 19 - Thomas Calculus

Question

In Exercises \(13-22,\) find the limit of each rational function (a) as \(x \rightarrow \infty\) and \((b)\) as \(x \rightarrow-\infty\) . $$g(x)=\frac{10 x^{5}+x^{4}+31}{x^{6}}$$

Step-by-Step Solution

Answer

The limit of \( g(x) \) as \( x \to \infty \) and \( x \to -\infty \) is 0.

Step 1: Identify Highest Degree Terms

In the rational function \( g(x) = \frac{10x^5 + x^4 + 31}{x^6} \), identify the highest degree terms in both the numerator and the denominator. The highest power term in the numerator is \(10x^5\) and in the denominator is \(x^6\).

Step 2: Divide Terms by Highest Power of Denominator

To simplify and find the limits, divide all terms in the numerator and the denominator by the highest power of \(x\) in the denominator, which is \(x^6\). This gives \( g(x) = \frac{10x^5/x^6 + x^4/x^6 + 31/x^6}{x^6/x^6} = \frac{10/x + 1/x^2 + 31/x^6}{1} \).

Step 3: Evaluate Limit as x Approaches Infinity

When \(x\) approaches infinity, each term involving \(x\) in the denominator approaches zero. Thus, \( \frac{10}{x} \rightarrow 0\), \( \frac{1}{x^2} \rightarrow 0\), and \( \frac{31}{x^6} \rightarrow 0\). Consequently, \( g(x) = 0 \).

Step 4: Evaluate Limit as x Approaches Negative Infinity

Similarly, as \(x\) approaches negative infinity, each term with \(x\) in the denominator still approaches zero: \( \frac{10}{x} \rightarrow 0 \), \( \frac{1}{x^2} \rightarrow 0 \), and \( \frac{31}{x^6} \rightarrow 0 \). Therefore, \( g(x) = 0 \) as well.

Key Concepts

Highest Degree Terms

When handling limits of rational functions, one important concept is identifying the *highest degree terms*. In the given function \( g(x) = \frac{10x^5 + x^4 + 31}{x^6} \), this means spotting the terms with the largest exponents. For the numerator, this is \(10x^5\), and for the denominator, it's \(x^6\). These terms are crucial as they typically dominate the behavior of the function, especially as \(x\) approaches infinity or negative infinity. By focusing on these leading terms, we can simplify and predict the function's behavior more efficiently, making the process much more comfortable to visualize and calculate.

Dividing Terms in Limits

After identifying the highest degree terms, the next step is *dividing all terms by the highest power* of the denominator. This technique is used to evaluate the limit. In our function, the highest term in the denominator is \(x^6\), so we divide every term in the function by \(x^6\). This simplifies \(g(x)\) to \(\frac{10/x + 1/x^2 + 31/x^6}{1}\). By doing this division, we can more easily analyze how each term behaves as \(x\) either grows very large or becomes very negative. This division often results in a simpler expression with terms that clearly tend to zero as \(x\) approaches infinity or negative infinity.

Behavior at Infinity

Understanding a function's *behavior at infinity* is crucial for evaluating limits. In our exercise, as \(x\) approaches infinity, the terms \(\frac{10}{x}\), \(\frac{1}{x^2}\), and \(\frac{31}{x^6}\) all tend to zero because the denominator becomes much larger than the numerator for each term. Thus, \(g(x) = 0\) when \(x\) approaches infinity. The same conclusion holds when \(x\) goes towards negative infinity. This is because the denominators in each fraction still grow much faster than the numerators regardless of whether \(x\) is positive or negative. Thus, knowing how individual terms behave allows us to determine the overall limit of the function as \(x\) approaches these extreme values.