Question

question
find all horizontal asymptotes of the following function.
f(x)=\frac{3 x-2}{(x+5)(3 x-2)}
answer attempt 1 out of 2
one horizontal asymptote
no horizontal asymptotes
one horizontal asymptote
two horizontal asymptotes

Explanation:

Step1: Simplify the function

Cancel out the common factor \(3x - 2\) (assuming \(x
eq\frac{2}{3}\)) in the numerator and denominator.
\(f(x)=\frac{3x - 2}{(x + 5)(3x - 2)}=\frac{1}{x + 5}\), \(x
eq\frac{2}{3}\)

Step2: Find the limit as \(x\to\pm\infty\)

Use the formula \(\lim_{x\to\pm\infty}\frac{a}{x + b}=0\) (where \(a = 1\), \(b=5\))
\(\lim_{x\to\infty}\frac{1}{x + 5}=0\) and \(\lim_{x\to-\infty}\frac{1}{x + 5}=0\)

Answer:

One Horizontal Asymptote (\(y = 0\))