Question

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=7xe^{-0.5x} ).
find any horizontal asymptotes of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
a. the function has one horizontal asymptote,
(type an equation.)
b. the function has two horizontal asymptotes. the top asymptote is and the bottom asymptote is
(type equations.)
c. there are no horizontal asymptotes.
find any vertical asymptotes of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
a. the function has one vertical asymptote,
(type an equation.)
b. the function has two vertical asymptotes. the leftmost asymptote is and the rightmost asymptote is
(type equations.)
c. there are no vertical asymptotes.

Explanation:

Step1: Find horizontal asymptotes

Recall the formula for horizontal asymptotes: $\lim_{x
ightarrow\infty}f(x)$ and $\lim_{x
ightarrow-\infty}f(x)$.
For $\lim_{x
ightarrow\infty}7xe^{-0.5x}$, use L - H rule (since it is in the $\infty\times0$ form. Rewrite as $\lim_{x
ightarrow\infty}\frac{7x}{e^{0.5x}}$.
Differentiate numerator and denominator: $\lim_{x
ightarrow\infty}\frac{7}{0.5e^{0.5x}} = 0$.
For $\lim_{x
ightarrow-\infty}7xe^{-0.5x}$, as $x
ightarrow-\infty$, $e^{-0.5x}=e^{|0.5x|}
ightarrow\infty$ and $x
ightarrow-\infty$, so $7xe^{-0.5x}
ightarrow-\infty$.

Step2: Find vertical asymptotes

The function $y = 7xe^{-0.5x}=\frac{7x}{e^{0.5x}}$ is a combination of a polynomial ($y = 7x$) and an exponential function ($y = e^{0.5x}$). The domain of $y = 7xe^{-0.5x}$ is all real numbers ($x\in(-\infty,\infty)$) since $e^{0.5x}>0$ for all $x\in R$.

Answer:

For horizontal asymptotes: A. The function has one horizontal asymptote, $y = 0$.
For vertical asymptotes: C. There are no vertical asymptotes.