Question

what happens to the value of $f(x) = \log_{4} x$ as $x$ approaches 0 from the right?
what happens to the value of $f(x) = \log_{4} x$ as $x$ approaches $+\infty$?

Explanation:

Step1: Analyze right limit at 0

Recall properties of $\log_b x$ where $b>1$. As $x \to 0^+$, $\log_4 x$ decreases without bound.
$\lim_{x \to 0^+} \log_4 x = -\infty$

Step2: Analyze limit at $+\infty$

For $b>1$, $\log_b x$ increases without bound as $x$ grows.
$\lim_{x \to +\infty} \log_4 x = +\infty$

Answer:

1. As $x$ approaches 0 from the right, the value of $f(x)=\log_4 x$ approaches $-\infty$ (decreases without bound).
2. As $x$ approaches $+\infty$, the value of $f(x)=\log_4 x$ approaches $+\infty$ (increases without bound).