part 5 of 10 points: 0 of 1 summarize the pertinent information obtained by applying the graphing strategy…

Explanation:

Step1: Recall asymptote definitions

Vertical asymptotes occur where the function is undefined and the limit approaches infinity. Horizontal asymptotes are found by considering the limit as $x\to\pm\infty$.

Step2: Analyze vertical asymptotes

The function $f(x)=6xe^{-0.5x}=\frac{6x}{e^{0.5x}}$ is defined for all real - valued $x$ since the exponential function $e^{0.5x}>0$ for all $x\in R$ and the numerator $6x$ is a polynomial. So, $\lim_{x\to a}f(x)$ exists for all real $a$. Thus, there are no vertical asymptotes.

Step3: Analyze horizontal asymptotes

We find $\lim_{x\to\infty}6xe^{-0.5x}=\lim_{x\to\infty}\frac{6x}{e^{0.5x}}$. Using L'Hopital's rule (since we have the indeterminate form $\frac{\infty}{\infty}$), if $y = \frac{6x}{e^{0.5x}}$, then $y'=\frac{6}{0.5e^{0.5x}}$. As $x\to\infty$, $\lim_{x\to\infty}\frac{6}{0.5e^{0.5x}} = 0$. And $\lim_{x\to-\infty}6xe^{-0.5x}=-\infty$ since as $x\to-\infty$, $e^{-0.5x}\to\infty$ and $6x\to-\infty$. So, the horizontal asymptote is $y = 0$ as $x\to\infty$. But for the part of the question about vertical asymptotes:

Answer:

C. There are no vertical asymptotes.