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

which of the statements a through i about the function ( y = f(x) ) graphed here are true, and which are false?\na. the statement ( limlimits_{x\rightarrow2}f(x) ) does not exist is false.\nb. the statement ( limlimits_{x\rightarrow2}f(x)=2 ) is false.\nc. the statement ( limlimits_{x\rightarrow1}f(x) ) does not exist is true.\nd. the statement ( limlimits_{x\rightarrow c}f(x) ) exists at every point ( c ) in ( (-1,1) ) is true.\ne. the statement ( limlimits_{x\rightarrow c}f(x) ) exists at every point ( c ) in ( (1,3) ) is true.\nf. the statement ( f(1)=-3 ) is false.\ng. the statement ( f(1)=0 ) is true.\nh. the statement ( f(2)=3 ) is

which of the statements a through i about the function ( y = f(x) ) graphed here are true, and which are false?\na. the statement ( limlimits_{x\rightarrow2}f(x) ) does not exist is false.\nb. the statement ( limlimits_{x\rightarrow2}f(x)=2 ) is false.\nc. the statement ( limlimits_{x\rightarrow1}f(x) ) does not exist is true.\nd. the statement ( limlimits_{x\rightarrow c}f(x) ) exists at every point ( c ) in ( (-1,1) ) is true.\ne. the statement ( limlimits_{x\rightarrow c}f(x) ) exists at every point ( c ) in ( (1,3) ) is true.\nf. the statement ( f(1)=-3 ) is false.\ng. the statement ( f(1)=0 ) is true.\nh. the statement ( f(2)=3 ) is

Answer

Explanation:

Step1: Analyze the limit as (x\to2)

For the limit (\lim_{x\to2}f(x)), we look at the left - hand limit and the right - hand limit. As (x) approaches (2) from the left and the right, the function values approach the same value. So (\lim_{x\to2}f(x)) exists, making statement a false. And since the limit value is not (2), statement b is false.

Step2: Analyze the limit as (x\to1)

As (x) approaches (1) from the left ((x\to1^{-})), the function values approach a certain value and as (x) approaches (1) from the right ((x\to1^{+})), the function values approach a different value. So (\lim_{x\to1}f(x)) does not exist, making statement c true.

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

For any (c\in(-1,1)), as (x) approaches (c) from the left and the right, the function values approach the same value. So (\lim_{x\to c}f(x)) exists for (c\in(-1,1)), making statement d true.

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

For any (c\in(1,3)), as (x) approaches (c) from the left and the right, the function values approach the same value. So (\lim_{x\to c}f(x)) exists for (c\in(1,3)), making statement e true.

Step5: Analyze (f(1))

Looking at the graph, (f(1)) is the value of the function at (x = 1). From the graph, (f(1)=0), so statement f ((f(1)=-3)) is false and statement g ((f(1) = 0)) is true.

Step6: Analyze (f(2))

Looking at the graph, (f(2)) is not (3). The open - circle at (x = 2) does not represent the function value. The function value at (x=2) is not (3), so statement h is false.

Answer:

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