given $f(x)=\frac{|x - 5|}{x - 5}$, which one of the following is true? select one: a. $lim_{x\rightarrow5}f(…

given $f(x)=\frac{|x - 5|}{x - 5}$, which one of the following is true? select one: a. $lim_{x\rightarrow5}f(x)=1$ b. $f(5)=1$ c. $f(x)$ is continuous at $x = 5$ d. $lim_{x\rightarrow5^{-}}f(x)\neqlim_{x\rightarrow5^{+}}f(x)$

given $f(x)=\frac{|x - 5|}{x - 5}$, which one of the following is true? select one: a. $lim_{x\rightarrow5}f(x)=1$ b. $f(5)=1$ c. $f(x)$ is continuous at $x = 5$ d. $lim_{x\rightarrow5^{-}}f(x)\neqlim_{x\rightarrow5^{+}}f(x)$

Answer

Explanation:

Step1: Analyze the absolute - value function

Recall that (|x - 5|=\begin{cases}x - 5, & x\geq5\-(x - 5), & x<5\end{cases}).

Step2: Calculate the left - hand limit

For (x\to5^{-}) (left - hand limit), (|x - 5|=-(x - 5)). Then (\lim_{x\to5^{-}}\frac{|x - 5|}{x - 5}=\lim_{x\to5^{-}}\frac{-(x - 5)}{x - 5}=- 1).

Step3: Calculate the right - hand limit

For (x\to5^{+}) (right - hand limit), (|x - 5|=x - 5). Then (\lim_{x\to5^{+}}\frac{|x - 5|}{x - 5}=\lim_{x\to5^{+}}\frac{x - 5}{x - 5}=1).

Step4: Analyze the function value and continuity

The function (f(x)=\frac{|x - 5|}{x - 5}) is not defined at (x = 5) since the denominator is (0) at (x = 5). Also, since (\lim_{x\to5^{-}}f(x)=-1) and (\lim_{x\to5^{+}}f(x)=1), (\lim_{x\to5^{-}}f(x)\neq\lim_{x\to5^{+}}f(x)).

Answer:

D. (\lim_{x\to5^{-}}f(x)\neq\lim_{x\to5^{+}}f(x))