suppose f(x) is defined as shown below. a. use the continuity checklist to show that f is not continuous at…

suppose f(x) is defined as shown below. a. use the continuity checklist to show that f is not continuous at 3. b. is f continuous from the left or right at 3? c. state the interval(s) of continuity. f(x) = { x² + 3x if x>3; 4x if x≤3 } a. why is f not continuous at 3? a. lim f(x) does not exist. x→3 b. although lim f(x) exists, it does not equal f(3). x→3 c. f(3) is not defined. b. choose the correct answer below. a. f is continuous from the right at 3. b. f is continuous from the left at 3. c. f is not continuous from the left or the right at 3. c. what are the interval(s) of continuity? (simplify your answer. type your answer in interval notation. use a comma to separate answ

suppose f(x) is defined as shown below. a. use the continuity checklist to show that f is not continuous at 3. b. is f continuous from the left or right at 3? c. state the interval(s) of continuity. f(x) = { x² + 3x if x>3; 4x if x≤3 } a. why is f not continuous at 3? a. lim f(x) does not exist. x→3 b. although lim f(x) exists, it does not equal f(3). x→3 c. f(3) is not defined. b. choose the correct answer below. a. f is continuous from the right at 3. b. f is continuous from the left at 3. c. f is not continuous from the left or the right at 3. c. what are the interval(s) of continuity? (simplify your answer. type your answer in interval notation. use a comma to separate answ

Answer

Explanation:

Step1: Calculate left - hand limit

For (x\leq3), (f(x) = 4x). So, (\lim_{x\rightarrow3^{-}}f(x)=\lim_{x\rightarrow3^{-}}4x = 4\times3=12).

Step2: Calculate right - hand limit

For (x > 3), (f(x)=x^{2}+3x). So, (\lim_{x\rightarrow3^{+}}f(x)=\lim_{x\rightarrow3^{+}}(x^{2}+3x)=3^{2}+3\times3=9 + 9=18). Since (\lim_{x\rightarrow3^{-}}f(x)\neq\lim_{x\rightarrow3^{+}}f(x)), (\lim_{x\rightarrow3}f(x)) does not exist.

Step3: Check left - hand continuity

Since (\lim_{x\rightarrow3^{-}}f(x)=12) and (f(3)=4\times3 = 12) (because when (x = 3), we use (f(x)=4x) as (x\leq3)), (f(x)) is continuous from the left at (x = 3).

Step4: Find intervals of continuity

The function (y = 4x) is continuous for all (x\leq3) and the function (y=x^{2}+3x) is continuous for all (x>3). So the intervals of continuity are ((-\infty,3]) and ((3,\infty)), which can be written as ((-\infty,\infty)\setminus{3}).

Answer:

a. A. (\lim_{x\rightarrow3}f(x)) does not exist. b. B. (f) is continuous from the left at 3. c. ((-\infty,3]\cup(3,\infty))