summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of…
summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of f(x)=6xe^(-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.
Answer
Explanation:
Step1: Recall horizontal - asymptote definition
We find the limits as $x\to\pm\infty$. For $y = f(x)=6xe^{-0.5x}=\frac{6x}{e^{0.5x}}$.
Step2: Find the limit as $x\to+\infty$
Use L'Hopital's rule. $\lim_{x\to+\infty}\frac{6x}{e^{0.5x}}$. Since it is in the $\frac{\infty}{\infty}$ form, by L'Hopital's rule, $\lim_{x\to+\infty}\frac{6x}{e^{0.5x}}=\lim_{x\to+\infty}\frac{6}{0.5e^{0.5x}} = 0$.
Step3: Find the limit as $x\to-\infty$
As $x\to-\infty$, $e^{-0.5x}\to+\infty$ and $6x\to-\infty$. But $\lim_{x\to-\infty}6xe^{-0.5x}=-\infty$.
Answer:
A. The function has one horizontal asymptote, $y = 0$