Question

find the limits in a), b), and c) below for the function (f(x)=\frac{5x}{x - 6}). use (-infty) and (infty) when appropriate. a) select the correct choice below and fill in any answer boxes in your choice. (lim_{x
ightarrow6^{-}}f(x)=) (simplify your answer.) a. (-infty) b. the limit does not exist and is neither (-infty) nor (infty). b) select the correct choice below and fill in any answer boxes in your choice. (lim_{x
ightarrow6^{+}}f(x)=) (simplify your answer.) a. (infty) b. the limit does not exist and is neither (-infty) nor (infty).

Explanation:

Step1: Analyze the function for \(x

ightarrow6^{-}\)
The function is \(f(x)=\frac{5x}{x - 6}\). When \(x
ightarrow6^{-}\), the numerator \(5x
ightarrow30\) and the denominator \(x - 6
ightarrow0^{-}\). So \(\lim_{x
ightarrow6^{-}}f(x)=-\infty\).

Step2: Analyze the function for \(x

ightarrow6^{+}\)
When \(x
ightarrow6^{+}\), the numerator \(5x
ightarrow30\) and the denominator \(x - 6
ightarrow0^{+}\). So \(\lim_{x
ightarrow6^{+}}f(x)=\infty\).

Answer:

a) A. \(\lim_{x
ightarrow6^{-}}f(x)=-\infty\)
b) A. \(\lim_{x
ightarrow6^{+}}f(x)=\infty\)