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
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

Explanation:

Step1: Recall the definition of the limit

The limit of a function \(y = f(x)\) as \(x\to c\), \(\lim_{x\to c}f(x)=L\) if and only if \(\lim_{x\to c^{-}}f(x)=\lim_{x\to c^{+}}f(x) = L\)

Step2: Analyze the function for \(x\in(-1,1)\)

For any \(c\in(-1,1)\), the left - hand limit \(\lim_{x\to c^{-}}f(x)\) and the right - hand limit \(\lim_{x\to c^{+}}f(x)\) are equal.

Answer:

True