Question

describe the horizontal and vertical asymptotes of $f(x)=\log_{4}x$. (5 points)○ the function $f(x)$ has a horizontal asymptote at $y = 4$ and a vertical asymptote at $x = 0$.○ the function $f(x)$ has a horizontal asymptote at $y = 0$ and no vertical asymptote.○ the function $f(x)$ has a horizontal asymptote at $y = 0$ and a vertical asymptote at $x = 4$.○ the function $f(x)$ has a vertical asymptote at $x = 0$ and no horizontal asymptote.

Explanation:

Step1: Analyze vertical asymptote

For $f(x)=\log_4 x$, the domain is $x>0$. As $x\to0^+$, $f(x)\to-\infty$. So vertical asymptote at $x=0$.

Step2: Analyze horizontal asymptote

As $x\to+\infty$, $f(x)=\log_4 x\to+\infty$; there is no constant $y=k$ that $f(x)$ approaches. So no horizontal asymptote.

Answer:

The function $f(x)$ has a vertical asymptote at $x = 0$ and no horizontal asymptote.