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?\n\n a. the statement ( limlimits_{x\to 2}f(x) ) does not exist is false.\n b. the statement ( limlimits_{x\to 2}f(x) = 2 ) is false.\n c. the statement ( limlimits_{x\to 1}f(x) ) does not exist is true.\n d. the statement ( limlimits_{x\to c}f(x) ) exists at every point ( c ) in ( (-1,1) ) is true.\n e. the statement ( limlimits_{x\to c}f(x) ) exists at every point ( c ) in ( (1,3) ) is true.\n f. the statement ( f(1)=-3 ) is false.\n g. the statement ( f(1)=0 ) is true.\n h. 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?\n\n a. the statement ( limlimits_{x\to 2}f(x) ) does not exist is false.\n b. the statement ( limlimits_{x\to 2}f(x) = 2 ) is false.\n c. the statement ( limlimits_{x\to 1}f(x) ) does not exist is true.\n d. the statement ( limlimits_{x\to c}f(x) ) exists at every point ( c ) in ( (-1,1) ) is true.\n e. the statement ( limlimits_{x\to c}f(x) ) exists at every point ( c ) in ( (1,3) ) is true.\n f. the statement ( f(1)=-3 ) is false.\n g. the statement ( f(1)=0 ) is true.\n h. the statement ( f(2)=3 ) is

Answer

Explanation:

Step1: Analyze limit at (x = 2)

For (x\to2), the left - hand limit and the right - hand limit are different. The left - hand limit (approaching from the left side of (x = 2)) and the right - hand limit (approaching from the right side of (x = 2)) do not match. So, (\lim_{x\to2}f(x)) does not exist. So, statement (a) is false and statement (b) is false.

Step2: Analyze limit at (x = 1)

As (x\to1), the left - hand limit and the right - hand limit are different. So, (\lim_{x\to1}f(x)) does not exist. So, statement (c) is true.

Step3: Analyze limit existence in intervals

For (x\in(-1,1)), the function is well - behaved (no breaks, jumps, or holes in the graph in this open interval). So, (\lim_{x\to c}f(x)) exists for every (c\in(-1,1)). So, statement (d) is true. For (x\in(1,3)), at (x = 2) (which is in the interval ((1,3))), (\lim_{x\to2}f(x)) does not exist. So, statement (e) is false.

Step4: Analyze function values

Looking at the graph, when (x = 1), (f(1)=1\neq0). So, statement (g) is false. Also, (f(1)\neq - 3), so statement (f) is true. When (x = 2), (f(2)=-3\neq3). So, statement (h) is false.

Answer:

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