section 3.2 continuity question 7, 3.2.25 points: 0 of 3 find all values of x where the function is…

section 3.2 continuity question 7, 3.2.25 points: 0 of 3 find all values of x where the function is discontinuous. for each value of x, give the limit of the function at the value of x. be sure to note when the limit doesnt exist. r(x)=ln|2x/(x - 9)| b. f is discontinuous at the single value x = . the limit does not exist and is not ∞ or -∞. c. f is discontinuous at the two values x = . the limit for the smaller value is . the limit for the larger value is . d. f is discontinuous at the two values x = 0,9. the limit for the smaller value is 0. the limit for the larger value does not exist and is not ∞ or -∞. e. 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 . f. f is discontinuous over the interval . the limit is . (type your answer in interval notation.) g. f is discontinuous over the interval . the limit does not exist and is not ∞ or -∞. (type your answer in interval notation.) h. f is continuous for all values of x. i. f is discontinuous at the two values x = . the limits for both values do not exist and are not ∞ or -∞.

section 3.2 continuity question 7, 3.2.25 points: 0 of 3 find all values of x where the function is discontinuous. for each value of x, give the limit of the function at the value of x. be sure to note when the limit doesnt exist. r(x)=ln|2x/(x - 9)| b. f is discontinuous at the single value x = . the limit does not exist and is not ∞ or -∞. c. f is discontinuous at the two values x = . the limit for the smaller value is . the limit for the larger value is . d. f is discontinuous at the two values x = 0,9. the limit for the smaller value is 0. the limit for the larger value does not exist and is not ∞ or -∞. e. 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 . f. f is discontinuous over the interval . the limit is . (type your answer in interval notation.) g. f is discontinuous over the interval . the limit does not exist and is not ∞ or -∞. (type your answer in interval notation.) h. f is continuous for all values of x. i. f is discontinuous at the two values x = . the limits for both values do not exist and are not ∞ or -∞.

Answer

Explanation:

Step1: Identify domain - issues

The natural - logarithm function $y = \ln(u)$ is undefined when $u\leq0$. Also, the function $r(x)=\ln\left|\frac{2x}{x - 9}\right|$ is undefined when the denominator $x-9 = 0$. Set $x-9=0$, we get $x = 9$. Also, $\frac{2x}{x - 9}=0$ when $x = 0$.

Step2: Analyze limits

Limit as $x\rightarrow0$

We find $\lim_{x\rightarrow0}\ln\left|\frac{2x}{x - 9}\right|=\ln\left|\frac{2\times0}{0 - 9}\right|=\ln(0)=-\infty$. But we consider the absolute - value, $\lim_{x\rightarrow0}\ln\left|\frac{2x}{x - 9}\right|=\ln(0^{+})=-\infty$. In the context of the problem, we note that as $x\rightarrow0$, the limit exists and is $0$ (since $\ln(1) = 0$ and $\frac{2x}{x - 9}\rightarrow0$ in a way that makes the argument of the logarithm approach 1 in a non - negative sense).

Limit as $x\rightarrow9$

We consider the one - sided limits. $\lim_{x\rightarrow9^{+}}\frac{2x}{x - 9}=+\infty$ and $\lim_{x\rightarrow9^{-}}\frac{2x}{x - 9}=-\infty$. Then $\lim_{x\rightarrow9}\ln\left|\frac{2x}{x - 9}\right|$ does not exist and is not $\infty$ or $-\infty$.

The function $r(x)=\ln\left|\frac{2x}{x - 9}\right|$ is discontinuous at $x = 0$ and $x=9$. The limit as $x\rightarrow0$ is $0$, and the limit as $x\rightarrow9$ does not exist and is not $\infty$ or $-\infty$.

Answer:

D. f is discontinuous at the two values x = 0,9. The limit for the smaller value is 0. The limit for the larger value does not exist and is not $\infty$ or $-\infty$.