(a) graph the given function, (b) find all values of x where the function is discontinuous, and (c) find the…

(a) graph the given function, (b) find all values of x where the function is discontinuous, and (c) find the limit from the left and the right at any values of x where the function is discontinuous. \n g(x)=\begin{cases}5&\text{if }x < - 3\\x^{2}+4&\text{if }-3leq xleq1\\5&\text{if }x > 1end{cases} \n(b) select the correct choice below and, if necessary, fill in the answer box to complete your choice. \na. the function is discontinuous at (x = - 3) (use a comma to separate answers as needed) \nb. the function is continuous for all values of x. \n(c) find the limit or limits from the left and from the right for any values of x found in part (b). select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. \na. the limit(s) from the left is/are. the limit(s) from the right is/are (use a comma to separate answers as needed) \nb. no discontinuities were found in part (b)

(a) graph the given function, (b) find all values of x where the function is discontinuous, and (c) find the limit from the left and the right at any values of x where the function is discontinuous. \n g(x)=\begin{cases}5&\text{if }x < - 3\\x^{2}+4&\text{if }-3leq xleq1\\5&\text{if }x > 1end{cases} \n(b) select the correct choice below and, if necessary, fill in the answer box to complete your choice. \na. the function is discontinuous at (x = - 3) (use a comma to separate answers as needed) \nb. the function is continuous for all values of x. \n(c) find the limit or limits from the left and from the right for any values of x found in part (b). select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. \na. the limit(s) from the left is/are. the limit(s) from the right is/are (use a comma to separate answers as needed) \nb. no discontinuities were found in part (b)

Answer

Explanation:

Step1: Check continuity at (x = - 3)

Left - hand limit: (\lim_{x\rightarrow - 3^{-}}g(x)=\lim_{x\rightarrow - 3^{-}}5 = 5) Right - hand limit: (\lim_{x\rightarrow - 3^{+}}g(x)=\lim_{x\rightarrow - 3^{+}}(x^{2}+4)=(-3)^{2}+4=9 + 4=13) Since (\lim_{x\rightarrow - 3^{-}}g(x)\neq\lim_{x\rightarrow - 3^{+}}g(x)), the function is discontinuous at (x=-3).

Step2: Check continuity at (x = 1)

Left - hand limit: (\lim_{x\rightarrow 1^{-}}g(x)=\lim_{x\rightarrow 1^{-}}(x^{2}+4)=1^{2}+4=5) Right - hand limit: (\lim_{x\rightarrow 1^{+}}g(x)=\lim_{x\rightarrow 1^{+}}5 = 5) Since (\lim_{x\rightarrow 1^{-}}g(x)=\lim_{x\rightarrow 1^{+}}g(x)), the function is continuous at (x = 1).

Step3: Find left - hand and right - hand limits at (x=-3)

Left - hand limit at (x=-3): (\lim_{x\rightarrow - 3^{-}}g(x)=5) Right - hand limit at (x=-3): (\lim_{x\rightarrow - 3^{+}}g(x)=(-3)^{2}+4 = 13)

Answer:

(b) A. The function is discontinuous at (x=-3) (c) A. The limit(s) from the left is/are (5). The limit(s) from the right is/are (13)