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

Explanation:

Step1: Calculate left - hand limit

For (x\lt1), (f(x) = 2x). So, (\lim_{x\rightarrow1^{-}}f(x)=\lim_{x\rightarrow1^{-}}2x = 2\times1=2).

Step2: Calculate right - hand limit

For (x\geq1), (f(x)=x^{2}+3x). So, (\lim_{x\rightarrow1^{+}}f(x)=\lim_{x\rightarrow1^{+}}(x^{2}+3x)=1^{2}+3\times1=1 + 3=4).

Step3: Check limit existence

Since (\lim_{x\rightarrow1^{-}}f(x)=2) and (\lim_{x\rightarrow1^{+}}f(x)=4), (\lim_{x\rightarrow1}f(x)) does not exist because the left - hand limit and the right - hand limit are not equal.

Step4: Analyze continuity from left and right

For continuity from the left, (\lim_{x\rightarrow1^{-}}f(x) = 2) and (f(1)=1^{2}+3\times1 = 4). Since (\lim_{x\rightarrow1^{-}}f(x)\neq f(1)), it is not continuous from the left. For continuity from the right, (\lim_{x\rightarrow1^{+}}f(x)=4) and (f(1) = 4), so (f(x)) is continuous from the right at (x = 1).

Step5: Find intervals of continuity

The function (y = 2x) is continuous for (x\lt1) and (y=x^{2}+3x) is continuous for (x\geq1). So the intervals of continuity are ((-\infty,1)) and ([1,\infty)), or in other words, ((-\infty,\infty)) (because it is continuous from the right at (x = 1)).

Answer:

a. B. (\lim_{x\rightarrow1}f(x)) does not exist. b. Continuous from the right at 1. c. ((-\infty,\infty))