Question

for each function, determine the long run behavior. question help: video

Explanation:

Step1: Recall the rule for horizontal asymptote

For a rational function \(y = \frac{f(x)}{g(x)}=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}\), if \(n=m\), the horizontal asymptote is \(y=\frac{a_n}{b_m}\).
For the function \(y=\frac{x^2 + 1}{x^2+2}\), here \(n = m=2\), \(a_n = 1\), \(b_m=1\).

Step2: Calculate the horizontal asymptote

Using the formula \(y=\frac{a_n}{b_m}\), we substitute \(a_n = 1\) and \(b_m = 1\). So \(y=\frac{1}{1}=1\).

Answer:

has a horizontal asymptote at \(y = 1\)