Question

Use the equation: ….g. For any vertical asymptote, describe the behavior (i.e, does the function approach positive infinity?) as the function approaches the asymptote(s) from the left and right. Ensure your work provides justification for your claims.(i only need help with question g)

Answer 1

$h(x)=\frac{x+7}{x^2-49}$

g. The vertical asymptote is located at x = 7, so, let's find the one-sided limits:

$\begin{gathered} \lim _{x\to7^-}(h(x))=\lim _{x\to7^-}\frac{7+7}{x^2-49}=14\lim _{x\to7^-}\frac{1}{x^2-49} \\ so\colon \\ x^2-49<0_{\text{ }}x\in(-\infty,7) \\ so\colon \\ 14\lim _{x\to7^-}\frac{1}{x^2-49}=14(-\infty)=-\infty \end{gathered}$$\begin{gathered} \lim _{x\to7^+}(h(x))=\lim _{x\to7^+}\frac{7+7}{x^2-49}=14\lim _{x\to7^+}\frac{1}{x^2-49} \\ so\colon \\ x^2-49>0_{\text{ }}x\in(7,\infty) \\ so\colon \\ 14\lim _{x\to7^+}\frac{1}{x^2-49}=14(\infty)=\infty \end{gathered}$

Therefore, when h(x) approaches to 7 from the left it tends to - infinity and when approaches to 7 from the right, it tends to +infinity