Summarize the pertinent information oblained by applying the graphing strategy and sketch the graph of $$ f(x)=7 x e^{-0.5 x} $$ A. The function has one horizontal asymptote, $$ y=0 $$. (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.

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To determine the horizontal and vertical asymptotes of the function $$ f(x) = 7xe^{-0.5x} $$, we need to analyze the behavior of the function as $$ x $$ approaches positive and negative infinity.

A. The function has one horizontal asymptote, $$ y=0 $$.
To find the horizontal asymptote, we need to determine the limit of the function as $$ x $$ approaches positive infinity.

The horizontal asymptote is $$ y=0 $$ because as $$ x $$ approaches positive infinity, the term $$ e^{-0.5x} $$ approaches 0, and the term $$ 7x $$ approaches infinity. Therefore, the horizontal asymptote is $$ y=0 $$.

B. The function has two horizontal asymptotes. The top asymptote is $$ y=0 $$ and the bottom asymptote is $$ y=0 $$.
Since the function has only one horizontal asymptote, which is $$ y=0 $$, there are no two horizontal asymptotes.

C. There are no horizontal asymptotes.
Since we have already determined that the function has one horizontal asymptote, $$ y=0 $$, there are no horizontal asymptotes.

Now, let's find the vertical asymptotes of the function $$ f(x) = 7xe^{-0.5x} $$.

A. The function has one vertical asymptote, $$ x=0 $$.
To find the vertical asymptote, we need to determine the value of $$ x $$ for which the function is undefined.

The function is undefined when the denominator of the function is equal to zero. In this case, the denominator is $$ e^{-0.5x} $$, which is never equal to zero. Therefore, there are no vertical asymptotes.

B. The function has two vertical asymptotes. The leftmost asymptote is $$ x=0 $$ and the rightmost asymptote is $$ x=0 $$.
Since we have already determined that there are no vertical asymptotes, there are no two vertical asymptotes.

C. There are no vertical asymptotes.
Since we have already determined that there are no vertical asymptotes, the correct answer is C. There are no vertical asymptotes.
The function $$ f(x) = 7xe^{-0.5x} $$ has one horizontal asymptote at $$ y=0 $$ and no vertical asymptotes.

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