on 3.2 continuity\nfind all values x = a where the function is discontinuous. for each value of x, give the…

on 3.2 continuity\nfind all values x = a where the function is discontinuous. for each value of x, give the limit of the function as x approaches a. be sure to note when the limit doesnt exist.\np(x)=|x - 8|/(x - 8)\nselect the correct choice below and, if necessary, fill in the answer box(es) within your choice. (use a comma to separate answers as needed.)\na. f is discontinuous at the single value x = . the limit is .\nb. f is discontinuous at the single value x = . the limit does not exist and is not ∞ or -∞.\nc. f is discontinuous at the two values x = . the limit for the smaller value is . the limit for the larger value is .\nd. f is discontinuous at the two values x = . the limit for the smaller value is . the limit for the larger value does not exist and is not ∞ or -∞.\ne. f is discontinuous at the two values x = . the limit for the smaller value does not exist and is not ∞ or -∞. the limit for the larger value is .\nf. f is discontinuous over the interval . the limit is . (type your answer in interval notation.)\ng. f is discontinuous over the interval . the limit does not exist and is not ∞ or -∞. (type your answer in interval notation.)

on 3.2 continuity\nfind all values x = a where the function is discontinuous. for each value of x, give the limit of the function as x approaches a. be sure to note when the limit doesnt exist.\np(x)=|x - 8|/(x - 8)\nselect the correct choice below and, if necessary, fill in the answer box(es) within your choice. (use a comma to separate answers as needed.)\na. f is discontinuous at the single value x = . the limit is .\nb. f is discontinuous at the single value x = . the limit does not exist and is not ∞ or -∞.\nc. f is discontinuous at the two values x = . the limit for the smaller value is . the limit for the larger value is .\nd. f is discontinuous at the two values x = . the limit for the smaller value is . the limit for the larger value does not exist and is not ∞ or -∞.\ne. f is discontinuous at the two values x = . the limit for the smaller value does not exist and is not ∞ or -∞. the limit for the larger value is .\nf. f is discontinuous over the interval . the limit is . (type your answer in interval notation.)\ng. f is discontinuous over the interval . the limit does not exist and is not ∞ or -∞. (type your answer in interval notation.)

Answer

Explanation:

Step1: Identify the point of discontinuity

A rational - function is discontinuous where the denominator is zero. For $p(x)=\frac{|x - 8|}{x - 8}$, set the denominator $x−8 = 0$. Solving $x−8 = 0$ gives $x = 8$.

Step2: Analyze the left - hand and right - hand limits

When $x\lt8$, $|x - 8|=-(x - 8)$. Then $\lim_{x\rightarrow8^{-}}\frac{|x - 8|}{x - 8}=\lim_{x\rightarrow8^{-}}\frac{-(x - 8)}{x - 8}=-1$. When $x\gt8$, $|x - 8|=x - 8$. Then $\lim_{x\rightarrow8^{+}}\frac{|x - 8|}{x - 8}=\lim_{x\rightarrow8^{+}}\frac{x - 8}{x - 8}=1$. Since $\lim_{x\rightarrow8^{-}}\frac{|x - 8|}{x - 8}\neq\lim_{x\rightarrow8^{+}}\frac{|x - 8|}{x - 8}$, the limit $\lim_{x\rightarrow8}\frac{|x - 8|}{x - 8}$ does not exist.

Answer:

B. f is discontinuous at the single value $x = 8$. The limit does not exist and is not $\infty$ or $-\infty$.