which of the statements a through i about the function ( y = f(x) ) graphed here are true, and which are…
which of the statements a through i about the function ( y = f(x) ) graphed here are true, and which are false?\n\n a. the statement ( lim_{x \to 2} f(x) ) does not exist is false.\n\n b. the statement ( lim_{x \to 2} f(x) = 2 ) is false.\n\n c. the statement ( lim_{x \to 1} f(x) ) does not exist is true.\n\n d. the statement ( lim_{x \to c} f(x) ) exists at every point ( c ) in ( (-1,1) ) is true.\n\n e. the statement ( lim_{x \to c} f(x) ) exists at every point ( c ) in ( (1,3) ) is
Answer
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