Math Problem Statement

Use the graph of the function f shown to estimate the indicated limits and the function value.

a. ModifyingBelow lim With font size decreased by 1 x right arrow minus 1 Superscript minus f left parenthesis x right parenthesislimx→−1−f(x)

b. ModifyingBelow lim With font size decreased by 1 x right arrow minus 1 Superscript plus f left parenthesis x right parenthesislimx→−1+f(x)

c. ModifyingBelow lim With font size decreased by 1 x right arrow minus 1 f left parenthesis x right parenthesislimx→−1 f(x)

d. f left parenthesis negative 1 right parenthesisf(−1)

-224-224xy

A coordinate system has a horizontal x-axis labeled from negative 2 to 5 in increments of 1 and a vertical y-axis labeled from negative 2 to 5 in increments of 1. A graph has three branches. The first branch is a line segment that begins at an open circle at (1, 1) and rises from right to left, ending at (0, 2). The second branch is a horizontal ray that begins at (0, 2) and extends to the left. The third branch is a ray that begins at a closed circle at (1, 2) and rises from left to right passing through an open circle at (2, 3).

Question content area bottom

Part 1

a. Find the limit.

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.

ModifyingBelow lim With font size decreased by 1 x right arrow minus 1 Superscript minus f left parenthesis x right parenthesis equalslimx→−1−f(x)=enter your response here

​(Type an​ integer.)

B.

The limit does not exist.

Part 2

b. Find the limit.

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.

ModifyingBelow lim With font size decreased by 1 x right arrow minus 1 Superscript plus f left parenthesis x right parenthesislimx→−1+f(x)equals=enter your response here

​(Type an​ integer.)

B.

The limit does not exist.

Part 3

c. Find the limit.

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.

ModifyingBelow lim With font size decreased by 1 x right arrow minus 1 f left parenthesis x right parenthesis equalslimx→−1 f(x)=enter your response here

​(Type an​ integer.)

B.

The limit does not exist.

Part 4

d. Find the function value.

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

Solution

To approach this problem, we need to examine the graph and identify the values corresponding to the limits and function value at x=−1x = -1. Let's go step by step for each part:

a. lim⁡x→−1−f(x)\lim_{x \to -1^-} f(x)

This limit represents the value of the function as xx approaches −1-1 from the left (i.e., as xx gets closer to −1-1 from values smaller than −1-1).

From the description, the graph indicates that the function behaves as a ray (the second branch) which starts at (0,2)(0, 2) and extends leftward. The leftward behavior of this ray suggests that as xx approaches −1-1 from the left, the function value remains constant at 2.

Answer for a: lim⁡x→−1−f(x)=2\lim_{x \to -1^-} f(x) = 2

b. lim⁡x→−1+f(x)\lim_{x \to -1^+} f(x)

This limit represents the value of the function as xx approaches −1-1 from the right (i.e., as xx gets closer to −1-1 from values greater than −1-1).

From the graph description, as we move rightward toward −1-1, the function is part of a line segment starting from the open circle at (1,1)(1, 1) and rising to (0,2)(0, 2). The graph shows that the function does not approach any particular value as x→−1+x \to -1^+ from the right because the function has a break at this point.

Answer for b: lim⁡x→−1+f(x) does not exist\lim_{x \to -1^+} f(x) \text{ does not exist}

c. lim⁡x→−1f(x)\lim_{x \to -1} f(x)

This limit is the overall limit of the function as xx approaches −1-1 from both the left and right.

Since the limit from the left is 2, but the limit from the right does not exist, the overall limit at x=−1x = -1 does not exist. This is because both one-sided limits must agree for the two-sided limit to exist.

Answer for c: lim⁡x→−1f(x) does not exist\lim_{x \to -1} f(x) \text{ does not exist}

d. f(−1)f(-1)

To determine the function value at x=−1x = -1, we check the graph at x=−1x = -1. However, based on the description, we don't see any closed circle or specific point on the graph that corresponds to x=−1x = -1. This suggests that f(−1)f(-1) is undefined.

Answer for d: f(−1) does not existf(-1) \text{ does not exist}

Summary of Answers:

  • a. lim⁡x→−1−f(x)=2\lim_{x \to -1^-} f(x) = 2
  • b. lim⁡x→−1+f(x) does not exist\lim_{x \to -1^+} f(x) \text{ does not exist}
  • c. lim⁡x→−1f(x) does not exist\lim_{x \to -1} f(x) \text{ does not exist}
  • d. f(−1) does not existf(-1) \text{ does not exist}

Would you like further clarification on any part of the problem? Here are some related questions you might find useful:

  1. How can we determine if a limit exists from the graph of a function?
  2. What does it mean for a function to have a one-sided limit that doesn't exist?
  3. What is the difference between an open circle and a closed circle on a graph in terms of limits and function values?
  4. How do we handle piecewise functions when analyzing limits?
  5. What are the conditions for the existence of a two-sided limit?

Tip: When analyzing limits from a graph, always check both the left-hand and right-hand behavior as you approach the point of interest!

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Math Problem Analysis

Mathematical Concepts

Limits
Graph Analysis
Piecewise Functions

Formulas

-

Theorems

Limit Theorems
One-sided Limits

Suitable Grade Level

Grades 9-12

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