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

Explanation:

Step1: Analyze the limit as \(x\to2\)

For the limit \(\lim_{x\to2}f(x)\), we look at the left - hand limit and the right - hand limit. As \(x\) approaches \(2\) from the left and the right, the function values approach the same value. So \(\lim_{x\to2}f(x)\) exists.

Step2: Analyze the value of \(\lim_{x\to2}f(x)\)

By observing the graph, as \(x\) approaches \(2\), the function values approach \(1\) (not \(2\)).

Step3: Analyze the limit as \(x\to1\)

As \(x\) approaches \(1\) from the left, the function values approach a certain value, and as \(x\) approaches \(1\) from the right, the function values approach a different value. So \(\lim_{x\to1}f(x)\) does not exist.

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

In the open interval \((-1,1)\), the function is continuous (except at the endpoints which are not included in the interval). For a continuous function at a point \(c\) in an open interval, \(\lim_{x\to c}f(x)=f(c)\) exists.

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

In the open interval \((1,3)\), the function is continuous (except at the endpoints which are not included in the interval). For a continuous function at a point \(c\) in an open interval, \(\lim_{x\to c}f(x)=f(c)\) exists.

Answer:

e. True