Question

which of the statements a through f about the function ( y = f(x) ) graphed here are true, and which are false?
a. the statement ( lim_{x
ightarrow2}f(x) ) does not exist is false.
b. the statement ( lim_{x
ightarrow2}f(x)=2 ) is false.
c. the statement ( lim_{x
ightarrow1}f(x) ) does not exist is true.
d. the statement ( lim_{x
ightarrow c}f(x) ) exists at every point ( c ) in ( (-1,1) ) is true.
e. the statement ( lim_{x
ightarrow 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 the limit as \(x

ightarrow2\)
For the limit \(\lim_{x
ightarrow2}f(x)\), 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 \(\lim_{x
ightarrow2}f(x)=1\).

• Statement a: Since \(\lim_{x

ightarrow2}f(x) = 1\) (exists), the statement “\(\lim_{x
ightarrow2}f(x)\) does not exist” is false.

• Statement b: Since \(\lim_{x

ightarrow2}f(x)=1
eq2\), the statement “\(\lim_{x
ightarrow2}f(x)=2\)” is false.

Step2: Analyze the limit as \(x

ightarrow1\)
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 \(\lim_{x
ightarrow1}f(x)=1\) (exists). So statement c is false.

Step3: Analyze the limit as \(x

ightarrow 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 \(x = c\), \(\lim_{x
ightarrow c}f(x)=f(c)\). So the statement “\(\lim_{x
ightarrow c}f(x)\) exists at every point \(c\) in \((-1,1)\)” is true.

Step4: Analyze the limit as \(x

ightarrow c\) for \(c\in(1,3)\)
At \(x = 2\), \(\lim_{x
ightarrow2}f(x)=1\). For other points \(c\in(1,3)\) (assuming the graph is continuous in the non - hole regions), the function is continuous. So the statement “\(\lim_{x
ightarrow c}f(x)\) 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