Question

instructions: you are not allowed to consult any notes, books, ai, or the internet during this quiz. a scientific calculator (no graphing or text capabilities) is allowed on this quiz. show all of your work on each problem. good luck!

1. given the graph of \\(f(x)\\) below, find each of the following:

(a) \\(f(0) =\\)
(b) \\(f(6) =\\)
(c) \\(f(7) =\\)
(d) \\(\lim_{x \to 6^-} f(x) =\\)
(e) \\(\lim_{x \to 6^+} f(x) =\\)
(f) \\(\lim_{x \to 6} f(x) =\\)
(g) \\(\lim_{x \to 7} f(x) =\\)

1. is \\(f(x)\\) continuous at \\(x = 7\\)? justify your answer.

1. is \\(f(x)\\) continuous at \\(x = 2\\)? justify your answer.

Explanation:

Analyze the graph and find values for Question 1

Let's examine the grid and the key points of the function \(f(x)\) on the interval \([0, 8]\):

• At \(x = 0\): There is a solid point at \((0, 2)\). Thus, \(f(0) = 2\).
• At \(x = 6\):
• The curve from the left approaches an open circle at \((6, 4)\). Thus, the left-hand limit is \(\lim_{x \to 6^-} f(x) = 4\).
• The curve to the right starts at an open circle at \((6, 0)\). Thus, the right-hand limit is \(\lim_{x \to 6^+} f(x) = 0\).
• Since the left-hand and right-hand limits are not equal, the two-sided limit \(\lim_{x \to 6} f(x)\) does not exist (DNE).
• There is no solid point at \(x = 6\), so \(f(6)\) is undefined (DNE).
• At \(x = 7\):
• There is an open circle on the curve at \((7, 1.5)\).
• There is a solid point plotted at \((7, 2)\). Thus, \(f(7) = 2\).
• The curve approaches the open circle from both sides, so \(\lim_{x \to 7} f(x) = 1.5\).
• At \(x = 8\): There is a solid point at \((8, 4)\).

Let's fill in the subparts of Question 1:

• (a) \(f(0) = 2\)
• (b) \(f(6) =\) Undefined (or DNE)
• (c) \(f(7) = 2\)
• (d) \(\lim_{x \to 6^-} f(x) = 4\)
• (e) \(\lim_{x \to 6^+} f(x) = 0\)
• (f) \(\lim_{x \to 6} f(x) =\) DNE
• (g) \(\lim_{x \to 7} f(x) = 1.5\)

Evaluate continuity at x = 7 for Question 2

For a function \(f(x)\) to be continuous at \(x = c\), three conditions must be met:

1. \(f(c)\) is defined.
2. \(\lim_{x \to c} f(x)\) exists.
3. \(\lim_{x \to c} f(x) = f(c)\).

Let's check these conditions at \(x = 7\):

1. \(f(7) = 2\) (defined).
2. \(\lim_{x \to 7} f(x) = 1.5\) (exists).
3. Since \(\lim_{x \to 7} f(x) = 1.5

eq f(7) = 2\), the third condition is violated.

Therefore, \(f(x)\) is not continuous at \(x = 7\).

Evaluate continuity at x = 2 for Question 3

Let's check the continuity conditions at \(x = 2\):

1. Looking at the graph, at \(x = 2\), the curve is solid and unbroken. The value is \(f(2) = 2.5\).
2. The curve approaches the same value from both the left and the right: \(\lim_{x \to 2^-} f(x) = 2.5\) and \(\lim_{x \to 2^+} f(x) = 2.5\). Thus, \(\lim_{x \to 2} f(x) = 2.5\).
3. Since \(\lim_{x \to 2} f(x) = f(2) = 2.5\), all three conditions for continuity are satisfied.

Therefore, \(f(x)\) is continuous at \(x = 2\).

Answer:

Question 1

(a) \(f(0) = 2\)
(b) \(f(6) =\) Undefined (or DNE)
(c) \(f(7) = 2\)
(d) \(\lim_{x \to 6^-} f(x) = 4\)
(e) \(\lim_{x \to 6^+} f(x) = 0\)
(f) \(\lim_{x \to 6} f(x) =\) DNE
(g) \(\lim_{x \to 7} f(x) = 1.5\)

Question 2

No, \(f(x)\) is not continuous at \(x = 7\) because \(\lim_{x \to 7} f(x) = 1.5\), which does not equal the function value \(f(7) = 2\).

Question 3

Yes, \(f(x)\) is continuous at \(x = 2\) because \(\lim_{x \to 2} f(x) = f(2) = 2.5\).