Question

For each function, determine the long run behavior:


x^3+1 divided by x^2+1


x^2+1 divided by x^2+2

Answer 1

Answer:

The long-run behavior of the function $f(x)=\frac{x^3+1}{x^2+1}$ as $x\rightarrow -\infty$,  $f(x)\rightarrow -\infty$ and as $x\rightarrow +\infty$,  $f(x)\rightarrow +\infty$.

The long-run behavior of the function $g(x)=\frac{x^2+1}{x^2+2}$ as

$x\rightarrow \pm \infty$, $g(x)\rightarrow 1$.

Step-by-step explanation:

The long-run behavior of rational functions refers to what happens to the graph of the function as the x-values get big or small.

Suppose that $f(x)=\frac{P(x)}{Q(x)}$ is a rational function. Let $ax^n$ be the leading term of P(x) and $bx^m$ be the leading term of Q(x). Then the long run behavior of f(x) is the same as the long run behavior of the power function

$\frac{ax^n}{bx^m}=\frac{a}{b}x^{n-m}$

We have the following functions:

(a) $f(x)=\frac{x^3+1}{x^2+1}$

The leading term of the numerator is $x^3$ and the leading term of the denominator is $x^2$. So, the long run behavior of f(x) is given by

$f(x)=\frac{x^3+1}{x^2+1}=\frac{x^3}{x^2}=x$

This tells us that when the inputs get extreme (i.e.,$x\rightarrow \pm \infty$) the outputs will be

As $x\rightarrow -\infty$, $f(x)\rightarrow -\infty$ and as $x\rightarrow +\infty$, $f(x)\rightarrow +\infty$

(b) $g(x)=\frac{x^2+1}{x^2+2}$

We note that the leading term of the numerator is $x^2$ and the leading term of the denominator is $x^2$. So, the long run behavior of g(x) is given by

$g(x)=\frac{x^2+1}{x^2+2}=\frac{x^2}{x^2}=1$

This tells us that when the inputs get extreme (i.e.,$x\rightarrow \pm \infty$) the outputs will be about 1. This means that the graph of g(x) gets closer and closer to the line y = 1 as $x\rightarrow \pm \infty$.