Question

End behavior of rational functions Suppose $f(x)=\frac{p(x)}{q(x)}$ is a rational function, where $p(x)=a_m{x}^m+a_{m-1}{x}^{m-1}+\cdots +a_2{x}^2+a_{1}x+a_0$ $q(x)=b_n{x}^n+b_{n-1}{x}^{n-1}+\cdots +b_2{x}^2+b_{1}x+b_0,a_m\ne 0$ and $b_n\ne 0$. a. Prove that if $m=n,$ then $\underset{x\to \pm \mathrm{\infty }}{lim}f(x)=\frac{a_m}{b_n}$ b. Prove that if \(m

Short answer

Answer: When the degrees of the numerator and denominator are equal (i.e., $m=n$), the limit as $x\to \pm \mathrm{\infty }$ is equal to the ratio of the leading coefficients $\frac{a_m}{b_n}$.

Step-by-step solution

Dividing both numerator and denominator by $x^n$

Start by dividing both the numerator $p(x)$ and denominator $q(x)$ by the highest power of x in the denominator, which is $x^n$. This will create a new function $g(x)=\frac{\tilde{p}(x)}{\tilde{q}(x)}$, where: $$\tilde{p}(x)=\frac{a_m}{x^{n-m}}+\frac{a_{m-1}}{x^{n-m+1}}+\cdots +\frac{a_2}{x^{n-2}}+\frac{a_1}{x^{n-1}}+\frac{a_0}{x^n}$$ and $$\tilde{q}(x)=b_n+\frac{b_{n-1}}{x}+\frac{b_{n-2}}{x^2}+\cdots +\frac{b_2}{x^{n-2}}+\frac{b_1}{x^{n-1}}+\frac{b_0}{x^n}$$ Notice that as $x\to \pm \mathrm{\infty }$, all the terms in $\tilde{p}(x)$ and $\tilde{q}(x)$ with a power of x in the denominator will approach 0.

Find the end behavior for the case $m=n$

In this case, the degrees of the numerator and denominator are equal. We examine the limit as $x\to \pm \mathrm{\infty }$: $$\underset{x\to \pm \mathrm{\infty }}{lim}g(x)=\underset{x\to \pm \mathrm{\infty }}{lim}\frac{\tilde{p}(x)}{\tilde{q}(x)}$$ Since all the terms in $\tilde{p}(x)$ and $\tilde{q}(x)$ with a power of x in the denominator approach 0 when $x\to \pm \mathrm{\infty }$, we will only consider the first term of both $\tilde{p}(x)$ and $\tilde{q}(x)$: $$\underset{x\to \pm \mathrm{\infty }}{lim}g(x)=\frac{a_m}{b_n}$$ This proves the first part of the problem.

Find the end behavior for the case \(m

In this case, the degree of the numerator is less than the degree of the denominator. We examine the limit as $x\to \pm \mathrm{\infty }$: $$\underset{x\to \pm \mathrm{\infty }}{lim}g(x)=\underset{x\to \pm \mathrm{\infty }}{lim}\frac{\tilde{p}(x)}{\tilde{q}(x)}$$ Since all the terms in the numerator approach 0 when $x\to \pm \mathrm{\infty }$, we have: $$\underset{x\to \pm \mathrm{\infty }}{lim}g(x)=0$$ This proves the second part of the problem. In conclusion, the end behavior of a rational function can be determined by examining the degrees of its numerator and denominator. If the degrees are equal (i.e., $m=n$), the limit as $x\to \pm \mathrm{\infty }$ is equal to the ratio of the leading coefficients $\frac{a_m}{b_n}$. If the degree of the numerator is less than the degree of the denominator (i.e., $m<n$), the limit as $x\to \pm \mathrm{\infty }$ is 0.

Key concepts

Limits at Infinity

Understanding the 'limits at infinity' concept is fundamental when analyzing the end behavior of rational functions. In essence, limits at infinity refer to the value that a function approaches as the independent variable, usually denoted as x, heads towards positive or negative infinity.

Imagine you're on a road that stretches forever. As you move forward, the scenery (representing the function's value) changes. But after a certain point, no matter how far you go, the scenery might start to look the same (this constant scenery is akin to the limit at infinity). In mathematical terms, when we write $\underset{x\to \pm \mathrm{\infty }}{lim}f(x)$, we're asking what constant value f(x) settles into as x moves towards infinity or negative infinity.

For rational functions, which are ratios of polynomials, these limits help predict the function's behavior at extreme values of x. By simplifying the function and observing how the terms behave when x is very large, we can determine this limiting value, if it exists. Every term that has a variable in the denominator effectively becomes negligible, as any number divided by infinity is essentially zero. This simplification process is critical in finding the limit at infinity for rational functions.

Rational Function Analysis

The analysis of rational functions involves breaking down the behavior of the function into comprehensible parts. A rational function, expressed as $f(x)=\frac{p(x)}{q(x)}$, is made up of a numerator $p(x)$ and a denominator $q(x)$, each of which is a polynomial.

In order to analyze the end behavior of these functions, we concentrate on the leading term of both the numerator and denominator. This is because, as x grows very large or very small, the leading terms dominate the behavior of the function, rendering all lower-degree terms insignificant.

Here's an interesting fact: Mathematics often mirrors the real world in the idea that the 'big players' have the most influence. Similarly, in rational functions, those big players are the terms with the highest power of x. By examining these leading terms through processes like long division or factoring, we get a clearer picture of how the function behaves at extreme values of x; it helps us predict the function's long-term trends.

Leading Coefficient Test

The Leading Coefficient Test is a shortcut that can save a lot of time and effort when analyzing end behavior of polynomials and rational functions. It's like having a map of those roads we talked about before: Instead of walking for miles and miles, you can just check the map to know what's at the end.

The test is simple. For a polynomial, look at the highest power term - its degree and leading coefficient determine the function's end behavior. And for a rational function, compare the degrees of the numerator and denominator ($m$ and $n$).

Here are the rules to remember:

• If $m=n$, the ratio of the leading coefficients $\frac{a_m}{b_n}$ is the horizontal asymptote, which means it's where the function will settle as x approaches infinity.
• If $m<n$, the function has a horizontal asymptote at y=0, indicating the function will flatten out close to the x-axis as x gets larger or smaller.
• If $m>n$, the function has no horizontal asymptote and grows without bound, because the numerator increases faster than the denominator as x grows large.

This test serves as a rapid assessment tool and offers a straightforward way to deduce the long-range behavior of a function without detailed calculations. Just like our road analogy, where the leading coefficient is the final destination, the Leading Coefficient Test tells us whether we'll end up at sea level, atop a mountain, or soaring into space.