Question

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

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

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 \(x = 1\), \(f(1)=1
eq0\). So, statement \(g\) is false. Also, \(f(1)
eq - 3\), so statement \(f\) is true. When \(x = 2\), \(f(2)=-3
eq3\). So, statement \(h\) is false.

Answer:

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