Question

determine ( limlimits_{x \to infty} f(x) ) and ( limlimits_{x \to -infty} f(x) ) for the following function. then give the horizontal asymptotes of ( f ) (if any)

( f(x) = \frac{43x^6 + 3x^2}{19x^5 - 2x} )

...

select the correct choice below and, if necessary, fill in the answer box to complete your choice

a. ( limlimits_{x \to -infty} f(x) = -infty ) (simplify your answer)

b. the limit does not exist and is neither ( infty ) nor ( -infty )

identify the horizontal asymptotes. 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 using ( y ) as the variable )

b. the function has two horizontal asymptotes. the top asymptote is and the bottom asymptote is
(type equations using ( y ) as the variable )

c. the function has no horizontal asymptotes

Explanation:

Step1: Analyze the degrees of numerator and denominator

The numerator \(43x^6 + 3x^2\) has degree \(6\), and the denominator \(19x^5 - 2x\) has degree \(5\). Since the degree of the numerator (\(6\)) is greater than the degree of the denominator (\(5\)), we analyze the limits as \(x\to\pm\infty\) by looking at the leading terms.

Step2: Find \(\lim_{x\to\infty} f(x)\)

For \(x\to\infty\), the leading term of the numerator is \(43x^6\) and the leading term of the denominator is \(19x^5\). So, \(f(x)\approx\frac{43x^6}{19x^5}=\frac{43}{19}x\). As \(x\to\infty\), \(\frac{43}{19}x\to\infty\), so \(\lim_{x\to\infty} f(x)=\infty\).

Step3: Find \(\lim_{x\to-\infty} f(x)\)

For \(x\to-\infty\), the leading term of the numerator is \(43x^6\) (even degree, so positive) and the leading term of the denominator is \(19x^5\) (odd degree, so negative). So, \(f(x)\approx\frac{43x^6}{19x^5}=\frac{43}{19}x\). As \(x\to-\infty\), \(\frac{43}{19}x\to-\infty\), so \(\lim_{x\to-\infty} f(x)=-\infty\).

Step4: Determine horizontal asymptotes

A horizontal asymptote exists if \(\lim_{x\to\infty} f(x)\) or \(\lim_{x\to-\infty} f(x)\) is a finite number. Since both limits as \(x\to\infty\) and \(x\to-\infty\) are infinite (\(\infty\) and \(-\infty\) respectively), there are no horizontal asymptotes.

Answer:

For \(\lim_{x\to\infty} f(x)\), the limit is \(\infty\); for \(\lim_{x\to-\infty} f(x)\), the limit is \(-\infty\). The function has no horizontal asymptotes, so the answer for horizontal asymptotes is option C.