Q:

For each function, determine the long run behavior:x^3+1 divided by x^2+1x^2+1 divided by x^2+2

Accepted Solution

A:
Answer:The long-run behavior of the function [tex]f(x)=\frac{x^3+1}{x^2+1}[/tex] as [tex]x\rightarrow -\infty[/tex], Β [tex]f(x)\rightarrow -\infty[/tex] and as [tex]x\rightarrow +\infty[/tex], Β [tex]f(x)\rightarrow +\infty[/tex].The long-run behavior of the function [tex]g(x)=\frac{x^2+1}{x^2+2}[/tex] as[tex]x\rightarrow \pm \infty[/tex], [tex]g(x)\rightarrow 1[/tex].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 [tex]f(x)=\frac{P(x)}{Q(x)}[/tex] is a rational function. Let [tex]ax^n[/tex] be the leading term of P(x) and [tex]bx^m[/tex] 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 [tex]\frac{ax^n}{bx^m}=\frac{a}{b}x^{n-m}[/tex]We have the following functions:(a) [tex]f(x)=\frac{x^3+1}{x^2+1}[/tex]The leading term of the numerator is [tex]x^3[/tex] and the leading term of the denominator is [tex]x^2[/tex]. So, the long run behavior of f(x) is given by[tex]f(x)=\frac{x^3+1}{x^2+1}=\frac{x^3}{x^2}=x[/tex] This tells us that when the inputs get extreme (i.e.,[tex]x\rightarrow \pm \infty[/tex]) the outputs will be As [tex]x\rightarrow -\infty[/tex], [tex]f(x)\rightarrow -\infty[/tex] and as [tex]x\rightarrow +\infty[/tex], [tex]f(x)\rightarrow +\infty[/tex](b) [tex]g(x)=\frac{x^2+1}{x^2+2}[/tex]We note that the leading term of the numerator is [tex]x^2[/tex] and the leading term of the denominator is [tex]x^2[/tex]. So, the long run behavior of g(x) is given by [tex]g(x)=\frac{x^2+1}{x^2+2}=\frac{x^2}{x^2}=1[/tex]This tells us that when the inputs get extreme (i.e.,[tex]x\rightarrow \pm \infty[/tex]) the outputs will be about 1. This means that the graph of g(x) gets closer and closer to the line y = 1 as [tex]x\rightarrow \pm \infty[/tex].