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 ( lim_{x \to 2} f(x) ) does not exist is false.
b. the statement ( lim_{x \to 2} f(x)=2 ) is false.
c. the statement ( lim_{x \to 1} f(x) ) does not exist is true.
d. the statement ( lim_{x \to c} f(x) ) exists at every point ( c ) in ( (-1,1) ) is true.
e. the statement ( lim_{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 true.
i. the statement ( f(2)=1 ) is

Explanation:

Step1: Analyze limit at \(x = 2\)

For \(\lim_{x
ightarrow2}f(x)\), the left - hand limit and the right - hand limit are equal. So, \(\lim_{x
ightarrow2}f(x)\) exists.

Step2: Analyze \(\lim_{x

ightarrow2}f(x)\) value
From the graph, \(\lim_{x
ightarrow2}f(x)=1
eq2\)

Step3: Analyze limit at \(x = 1\)

At \(x = 1\), the left - hand limit (approaching from the left side of \(x = 1\)) and the right - hand limit (approaching from the right side of \(x = 1\)) are not equal. So, \(\lim_{x
ightarrow1}f(x)\) does not exist.

Step4: Analyze limit in \((-1,1)\)

In the open interval \((-1,1)\), the function is continuous (no breaks or jumps). So, \(\lim_{x
ightarrow c}f(x)\) exists for every \(c\in(-1,1)\)

Step5: Analyze limit in \((1,3)\)

In the open interval \((1,3)\), the function is continuous (no breaks or jumps). So, \(\lim_{x
ightarrow c}f(x)\) exists for every \(c\in(1,3)\)

Step6: Analyze \(f(1)\)

From the graph, \(f(1)=0
eq - 3\)

Step7: Analyze \(f(1)\)

From the graph, \(f(1) = 0\)

Step8: Analyze \(f(2)\)

From the graph, \(f(2)=1
eq3\)

Step9: Analyze \(f(2)\)

From the graph, \(f(2)=1\)

Answer:

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