Question

find the equation of all horizontal asymptotes of the following function.
f(x)=\frac{24}{5x}-6
answer attempt 1 out of 2
no horizontal asymptotes
no horizontal asymptotes
one horizontal asymptote
two horizontal asymptotes

Explanation:

Step1: Recall the rule for horizontal asymptotes

For a rational function \(y = \frac{a}{x}+b\) (where \(a
eq0\)), we find the limit as \(x\to\pm\infty\). The formula for horizontal asymptote is \(y = \lim_{x\to\pm\infty}f(x)\).

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

We know that \(\lim_{x\to\pm\infty}\frac{24}{5x}=0\) (since \(\lim_{x\to\pm\infty}\frac{c}{x}=0\) for any constant \(c\)). Then \(y=\lim_{x\to\pm\infty}(\frac{24}{5x}-6)\). Using the limit property \(\lim_{x\to a}(u(x)+v(x))=\lim_{x\to a}u(x)+\lim_{x\to a}v(x)\), we have \(y = 0 - 6\).

Answer:

One Horizontal Asymptote, \(y=-6\)