Question

determine \\( \lim\limits_{x \to \infty} f(x) \\) and \\( \lim\limits_{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

\\( \boldsymbol{\checkmark} \\) a. \\( \lim\limits_{x \to -\infty} f(x) = -\infty \\) (simplify your answer.)
\\( \circ \\) 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.
\\( \circ \\) a. the function has one horizontal asymptote, \\( \square \\) (type an equation using \\( y \\) as the variable.)
\\( \boldsymbol{\odot} \\) b. the function has two horizontal asymptotes. the top asymptote is \\( \square \\) and the bottom asymptote is \\( \square \\) (type equations using \\( y \\) as the variable.)
\\( \circ \\) c. the function has no horizontal asymptotes.

Explanation:

Step1: Analyze limit as \( x \to \infty \)

Divide numerator and denominator by \( x^5 \):
\( f(x)=\frac{43x + \frac{3}{x^3}}{19 - \frac{2}{x^4}} \). As \( x \to \infty \), \( \frac{3}{x^3} \to 0 \), \( \frac{2}{x^4} \to 0 \). So \( \lim_{x \to \infty} f(x)=\lim_{x \to \infty}\frac{43x}{19}=\infty \).

Step2: Analyze limit as \( x \to -\infty \)

Divide numerator and denominator by \( x^5 \) (note \( x^5 \) is negative as \( x \to -\infty \), so \( \frac{1}{x^5} \) is negative):
\( f(x)=\frac{43x + \frac{3}{x^3}}{19 - \frac{2}{x^4}} \). As \( x \to -\infty \), \( 43x \to -\infty \), others \( \to 0 \). So \( \lim_{x \to -\infty} f(x)=-\infty \).

Step3: Determine horizontal asymptotes

Horizontal asymptotes exist if \( \lim_{x \to \pm\infty} f(x) \) is finite. Here, both limits are infinite, so no horizontal asymptotes.

Answer:

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