Question

find the limits in a), b), and c) below for the function f(x) = $\frac{6x}{x - 8}$. use -∞ and ∞ when appropriate.
a) select the correct choice below and fill in any answer boxes in your choice.
a. $lim_{x
ightarrow8^{-}}f(x)=square$
(simplify your answer.)
b. the limit does not exist and is neither -∞ nor ∞.

Explanation:

Step1: Analyze left - hand limit

We want to find $\lim_{x
ightarrow8^{-}}\frac{6x}{x - 8}$. As $x
ightarrow8^{-}$, $x$ approaches 8 from the left - hand side. So, $x<8$, then $x - 8<0$, and $6x
ightarrow48$ as $x
ightarrow8$.

Step2: Determine the sign and value of the limit

Let $y=x - 8$, then $x=y + 8$ and the function becomes $\frac{6(y + 8)}{y}=\frac{6y+48}{y}=6+\frac{48}{y}$. As $x
ightarrow8^{-}$, $y
ightarrow0^{-}$. So, $\frac{48}{y}
ightarrow-\infty$. Then $6+\frac{48}{y}
ightarrow-\infty$.

Answer:

A. $\lim_{x
ightarrow8^{-}}f(x)=-\infty$