Question

In Exercises 43 and 44, which of the statements are true about the
function y = f(x) graphed there, and which are false?

Answer 1

A function is continuous at a point n, if we have;

1) $\lim\limits_{x \to n} f(x)$ is defined

2) $\lim\limits_{x \to n} f(x)$ exist, or has a value

3) $\lim\limits_{x \to n} f(x) = \mathbf{f(n)}$

The true statement is statement (a)

(a) True

Statements b, c, d, e, f, g, h, i, and j are false

(b) False, (c) False,  (d) False,  (e) False,  (f) False,  (g) False,  (h) False,  (i) False

The reason the above selection are correct is as follows:

The given parameters are;

The domains and ranges of the function are:

To the left: Domain; 1 ≤ x < 0, Range; 1 ≤ f(x) < 0

To the right: Domain; 0 < x < 1, Range; 0 < f(x) < 1

The function is discontinuous at x = 0, and x = 1

The statements are analyzed individually as follows:

(a) $\lim\limits_{x \to -1^+} f(x) = 1$ True

The reason the above statement is true is that a domain of the function is 1 ≤ x < 0, -1 is the left boundary of the function

Therefore as x tends to -1 from the right, which is -1⁺, the function is defined, which is f(-1) = 1

(b) $\lim\limits_{x \to 0^-} f(x) = 0$ False

An open hole on a graph, indicates that the function is discontinuous at the value of x = 0

Therefore, as x tends to 0, f(x) is not defined

(c)  $\lim\limits_{x \to 0^-} f(x) = 1$ False

The function is discontinuous at the value of x = 0

(d) $\lim\limits_{x \to 0^-} f(x) = \lim\limits_{x \to 0^+} f(x)$ False

There is a discontinuity at x = 0

(e) $\lim\limits_{x \to 0^-} f(x)$ exist False

At x = 0, the function is discontinuous

(f) $\lim\limits_{x \to 0} f(x) = 0$ False

At x = 0, the f(x) is discontinuous

(g) $\lim\limits_{x \to 0} f(x) = 1$ False

From the graph at x = 0 f(x) is discontinuous

(h) $\lim\limits_{x \to 1} f(x) = 1$ False

From the graph, $\lim\limits_{x \to 1} f(x)$ Does not exist

(i) $\lim\limits_{x \to 1} f(x) = 0$ False

$\lim\limits_{x \to 1} f(x)$ Does not exist

(j) $\lim\limits_{x \to 2^-} f(x) = 2$ False

The function is not defined at x = 2⁻

Lear more about continuous functions here:

https://brainly.com/question/18415249