use a graph or a table to find the following limit. lim log9(x) x→0+ lim log9(x)= (simplify your answer.) x→0+
use a graph or a table to find the following limit. lim log9(x) x→0+ lim log9(x)= (simplify your answer.) x→0+
Answer
Explanation:
Step1: Recall the property of logarithmic functions
The general form of a logarithmic function is $y = \log_a(x)$, and its graph has a vertical - asymptote at $x = 0$. As $x\to0^{+}$ for $y=\log_a(x)$ where $a>1$, the function value decreases without bound. For the function $y = \log_9(x)$, when $x$ approaches $0$ from the right - hand side. Let's consider the exponential form of the logarithm. If $y=\log_9(x)$, then $x = 9^y$. As $x\to0^{+}$, we need to find what $y$ approaches. We know that as $y\to-\infty$, $9^y=\frac{1}{9^{-y}}\to0$.
Step2: Determine the limit value
As $x$ approaches $0$ from the positive side for the function $y=\log_9(x)$, the value of $y$ approaches $-\infty$.
Answer:
$-\infty$