Question

1. (5 points) which of the following functions is not continuous at $x = 3$? there is only one correct answer. a. $\frac{x - 3}{x + 3}$ b. $\frac{x + 3}{x - 3}$ c. $ln(x + 3)$ d. $(3x - 9)^2$ e. none of the above

Explanation:

Step1: Recall continuity condition

A function $y = f(x)$ is continuous at $x = a$ if $\lim_{x
ightarrow a}f(x)=f(a)$ and the function is well - defined at $x = a$.

Step2: Analyze option A

For $f(x)=\frac{x - 3}{x + 3}$, when $x = 3$, $f(3)=\frac{3-3}{3 + 3}=0$. The function is well - defined and $\lim_{x
ightarrow3}\frac{x - 3}{x + 3}=0$.

Step3: Analyze option B

For $f(x)=\frac{x + 3}{x - 3}$, when $x = 3$, the denominator is $0$. So $f(3)$ is not defined. And $\lim_{x
ightarrow3}\frac{x + 3}{x - 3}$ approaches $\pm\infty$ depending on the side of approach. The function is not continuous at $x = 3$.

Step4: Analyze option C

For $f(x)=\ln(x + 3)$, when $x = 3$, $f(3)=\ln(3 + 3)=\ln6$. The function is well - defined and $\lim_{x
ightarrow3}\ln(x + 3)=\ln6$.

Step5: Analyze option D

For $f(x)=(3x - 9)^2$, when $x = 3$, $f(3)=(3\times3 - 9)^2=0$. The function is well - defined and $\lim_{x
ightarrow3}(3x - 9)^2=0$.

Answer:

B. $\frac{x + 3}{x - 3}$