Question

find the limits in a), b), and c) below for the function f(x) = 5x / (x - 6). use -∞ and ∞ when appropriate. a) select the correct choice below and fill in any answer boxes in your choice. a. lim f(x) = (simplify your answer.) x→6⁻ b. the limit does not exist and is neither -∞ nor ∞.

Explanation:

Step1: Analyze the left - hand limit

We want to find $\lim_{x
ightarrow6^{-}}\frac{5x}{x - 6}$. As $x
ightarrow6^{-}$, we consider values of $x$ that are less than 6 but getting closer to 6. Let $x=6 - h$, where $h>0$ and $h
ightarrow0$. Then $\frac{5x}{x - 6}=\frac{5(6 - h)}{(6 - h)-6}=\frac{30-5h}{-h}=\frac{-30 + 5h}{h}$.

Step2: Evaluate the limit

As $h
ightarrow0^{+}$, $\frac{-30 + 5h}{h}
ightarrow-\infty$ since the numerator approaches - 30 (a non - zero constant) and the denominator is a positive number approaching 0.

Answer:

A. $-\infty$