Question

find the limits in a), b), and c) below for the function f(x) = \frac{6x}{x - 8}. use -\infty and \infty when appropriate.
a) select the correct choice below and fill in any answer boxes in your choice.
a. \lim_{x\to8^{-}} f(x)=-\infty (simplify your answer.)
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.
a. \lim_{x\to8^{+}} f(x)=\infty (simplify your answer.)
b. the limit does not exist and is neither -\infty nor \infty.
c) select the correct choice below and fill in any answer boxes in your choice.
a. \lim_{x\to8} f(x)= (simplify your answer.)
b. the limit does not exist and is neither -\infty nor \infty.

Explanation:

Step1: Analyze left - hand limit as $x\to8^{-}$

When $x\to8^{-}$, we consider values of $x$ that are less than 8. For the function $f(x)=\frac{6x}{x - 8}$, as $x\to8^{-}$, the numerator $6x$ approaches $6\times8 = 48$ (a positive number), and the denominator $x - 8$ approaches 0 from the negative side. So, $\lim_{x\to8^{-}}\frac{6x}{x - 8}=-\infty$.

Step2: Analyze right - hand limit as $x\to8^{+}$

When $x\to8^{+}$, we consider values of $x$ that are greater than 8. The numerator $6x$ approaches $48$ (positive), and the denominator $x - 8$ approaches 0 from the positive side. So, $\lim_{x\to8^{+}}\frac{6x}{x - 8}=\infty$.

Step3: Analyze two - sided limit as $x\to8$

Since $\lim_{x\to8^{-}}f(x)=-\infty$ and $\lim_{x\to8^{+}}f(x)=\infty$, the two - sided limit $\lim_{x\to8}f(x)$ does not exist.

Answer:

a) A. $\lim_{x\to8^{-}}f(x)=-\infty$
b) A. $\lim_{x\to8^{+}}f(x)=\infty$
c) B. The limit does not exist and is neither $-\infty$ nor $\infty$.