Question

evaluate the following: lim_{x→4^{-}}\frac{|x - 4|}{x - 4}
-∞
∞
1
-1
\frac{0}{0}
question 5
1 pts
evaluate the following: lim_{x→0^{+}}\ln(x)
0
1
\frac{1}{0}
-∞
∞

Explanation:

Step1: Analyze left - hand limit of absolute value

When $x\to4^{-}$, $x - 4<0$, so $|x - 4|=-(x - 4)$. Then $\lim_{x\to4^{-}}\frac{|x - 4|}{x - 4}=\lim_{x\to4^{-}}\frac{-(x - 4)}{x - 4}$.

Step2: Simplify the expression

$\lim_{x\to4^{-}}\frac{-(x - 4)}{x - 4}=- 1$.

Step3: Analyze $\lim_{x\to0^{+}}\ln(x)$

The natural - logarithm function $y = \ln(x)$ is defined for $x>0$. As $x$ approaches $0$ from the right, the function $\ln(x)$ approaches negative infinity. That is, $\lim_{x\to0^{+}}\ln(x)=-\infty$.

Answer:

For the first limit $\lim_{x\to4^{-}}\frac{|x - 4|}{x - 4}$: D. $-1$
For the second limit $\lim_{x\to0^{+}}\ln(x)$: D. $-\infty$