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Chapter 2: Problem 52

Suppose \(p\) and \(q\) are polynomials and the horizonal axis is an asymptote of the graph of \(\frac{p}{q}\). Explain why $$ \operatorname{deg} p<\operatorname{deg} q $$

Short Answer

Expert verified
When the horizontal axis is an asymptote of the graph of a rational function \(\frac{p}{q}\), it means that the graph of the function approaches the horizontal axis as \(x\) approaches infinity or negative infinity. To ensure that the limit is 0, which corresponds to the horizontal axis being an asymptote, we must have the degree of the polynomial in the numerator (\(p\)) be less than the degree of the polynomial in the denominator (\(q\)). So, \(\operatorname{deg} p < \operatorname{deg} q\).

Step by step solution

01

Recall the definition of an asymptote and the behavior near asymptotes

The horizontal axis (or the x-axis) is an asymptote of the graph of the rational function \(\frac{p}{q}\) means that the graph of the function approaches the horizontal axis as \(x\) approaches infinity (\(x\rightarrow \infty\)) or negative infinity (\(x\rightarrow -\infty\)). Mathematically, this means that: \(\lim_{x \to \infty} \frac{p(x)}{q(x)} = 0\) and \(\lim_{x \to -\infty} \frac{p(x)}{q(x)} = 0\).
02

Examine the behavior of the rational function at infinity

To determine the behavior of the rational function \(\frac{p}{q}\) as \(x\) approaches infinity or negative infinity, we can apply the rule for finding limits of rational functions at infinity, which is: Given a rational function \(\frac{r(x)}{s(x)}\), where \(r(x)\) and \(s(x)\) are polynomials, then \(\lim_{x \to \infty} \frac{r(x)}{s(x)}\) or \(\lim_{x \to -\infty} \frac{r(x)}{s(x)}\) is determined by the relationship between the degrees of the polynomials \(r(x)\) and \(s(x)\).
03

Examine the three possible relationships between the degrees of the numerator and the denominator

There are three possible scenarios for the relationship between the degrees of the polynomials \(p(x)\) and \(q(x)\): 1. If \(\operatorname{deg} p < \operatorname{deg} q\), then the limit of the rational function will be 0. 2. If \(\operatorname{deg} p = \operatorname{deg} q\), then the limit of the rational function will be the ratio of the leading coefficients of \(p\) and \(q\). 3. If \(\operatorname{deg} p > \operatorname{deg} q\), then the limit of the rational function will be infinity or negative infinity, depending on the coefficients of the leading terms.
04

Apply the given condition and deduce the relationship between the degrees

Since the horizontal axis is an asymptote of the graph of \(\frac{p}{q}\), we know from step 1 that: \(\lim_{x \to \infty} \frac{p(x)}{q(x)} = 0\) and \(\lim_{x \to -\infty} \frac{p(x)}{q(x)} = 0\). Now, based on the three scenarios in step 3, the only scenario that will ensure the limit is 0 is when \(\operatorname{deg} p < \operatorname{deg} q\). Therefore, when the horizontal axis is an asymptote of the graph of \(\frac{p}{q}\), we must have \(\operatorname{deg} p < \operatorname{deg} q\).

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