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

Explanation:

Step1: Recall the definition of the limit

The limit of a function \( \lim_{x
ightarrow a}f(x) = L \) if and only if \( \lim_{x
ightarrow a^{-}}f(x)=\lim_{x
ightarrow a^{+}}f(x)=L \)

Step2: Analyze the left - hand and right - hand limits as \( x

ightarrow1 \)
As \( x \) approaches \( 1 \) from the left (\( x
ightarrow1^{-} \)), we observe the behavior of the function values. As \( x \) approaches \( 1 \) from the right (\( x
ightarrow1^{+} \)), we also observe the behavior of the function values.
From the graph, \( \lim_{x
ightarrow1^{-}}f(x)=1 \) and \( \lim_{x
ightarrow1^{+}}f(x)=1 \)

Step3: Determine the limit as \( x

ightarrow1 \)
Since \( \lim_{x
ightarrow1^{-}}f(x)=\lim_{x
ightarrow1^{+}}f(x) = 1 \), by the definition of the limit \( \lim_{x
ightarrow1}f(x)=1 \)

Answer:

The statement \( \lim_{x
ightarrow1}f(x) \) does not exist is false.