which of the statements a through f about the function ( y = f(x) ) graphed here are true, and which are…

Explanation:

Step1: Analyze the limit as (x\rightarrow2)

For the limit (\lim_{x\rightarrow2}f(x)), we look at the left - hand limit and the right - hand limit. As (x) approaches (2) from the left, (y = 1) (from the semi - circle part of the graph). As (x) approaches (2) from the right, (y = 1). So (\lim_{x\rightarrow2}f(x)=1).

• Statement a: Since (\lim_{x\rightarrow2}f(x) = 1) (exists), the statement “(\lim_{x\rightarrow2}f(x)) does not exist” is false.
• Statement b: Since (\lim_{x\rightarrow2}f(x)=1\neq2), the statement “(\lim_{x\rightarrow2}f(x)=2)” is false.

Step2: Analyze the limit as (x\rightarrow1)

As (x) approaches (1) from the left, (y) approaches (1) (from the left - hand side of the semi - circle). As (x) approaches (1) from the right, (y) approaches (1) (from the right - hand side of the semi - circle). So (\lim_{x\rightarrow1}f(x)=1) (exists). So statement c is false.

Step3: Analyze the limit as (x\rightarrow c) for (c\in(-1,1))

For any (c\in(-1,1)), the function is continuous (assuming the graph is continuous in this interval). For a continuous function (y = f(x)) at (x = c), (\lim_{x\rightarrow c}f(x)=f(c)). So the statement “(\lim_{x\rightarrow c}f(x)) exists at every point (c) in ((-1,1))” is true.

Step4: Analyze the limit as (x\rightarrow c) for (c\in(1,3))

At (x = 2), (\lim_{x\rightarrow2}f(x)=1). For other points (c\in(1,3)) (assuming the graph is continuous in the non - hole regions), the function is continuous. So the statement “(\lim_{x\rightarrow c}f(x)) exists at every point (c) in ((1,3))” is true.

Step5: Analyze (f(1))

Looking at the graph, when (x = 1), (y = 1). So the statement (f(1)=-3) is false and the statement (f(1) = 0) is false.

Step6: Analyze (f(2))

Looking at the graph, when (x = 2), (y = 1). So the statement (f(2)=3) is false.

Answer:

a. False b. False c. False d. True e. True f. False g. False h. False