Math Problem Statement

Use the graph of the function f shown to estimate the following limits and the function value. Complete parts​ (A) through​ (D). -5 5 -5 5 x y

A coordinate system has a horizontal x-axis labeled from negative 5 to 5 in increments of 1 and a vertical y-axis labeled from negative 5 to 5 in increments of 1. A graph consists of two curves, two line segments, and a point. The first line segment extends between open circles at (negative 1, 3) and (0, 2). The second line segment extends from the open circle at (0, 2) to a closed circle at (1, 3). The first curve begins at a closed circle at (negative 1, negative 3) and rises right to left at an increasing rate passing through an open circle at (negative 2, negative 2) and through (negative 3, 1). The second curve begins at an open circle at (1, negative 3) and rises left to right at an increasing rate passing through (3, 1). A solid circle is plotted at (negative 2, 0). Question content area bottom Part 1 ​(A) Find the limit ModifyingBelow lim With font size decreased by 1 x right arrow minus 2 Superscript minus f left parenthesis x right parenthesis. 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 2 Superscript minus f left parenthesis x right parenthesis equals    enter your response here ​(Type an integer or a​ decimal.) B. The limit does not exist. Part 2 ​(B) Find the limit ModifyingBelow lim With font size decreased by 1 x right arrow minus 2 Superscript plus f left parenthesis x right parenthesis. 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 2 Superscript plus f left parenthesis x right parenthesis equals    enter your response here ​(Type an integer or a​ decimal.) B. The limit does not exist. Part 3 ​(C) Find the limit ModifyingBelow lim With font size decreased by 1 x right arrow minus 2 f left parenthesis x right parenthesis. 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 2 f left parenthesis x right parenthesis equals    enter your response here ​(Type an integer or a​ decimal.) B. The limit does not exist. Part 4 ​(D) Find the function value f left parenthesis negative 2 right parenthesis. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. f left parenthesis negative 2 right parenthesis equals    enter your response here ​(Type an integer or a​ decimal.) B. The function is not defined at xequalsnegative 2.

Solution

Let's analyze the graph and address each part of the question. Here's how we approach the problem:

Part 1:

Find the limit lim⁡x→−2−f(x)\lim_{{x \to -2^-}} f(x):

  • This asks for the limit of the function f(x)f(x) as xx approaches −2-2 from the left side (negative direction).
  • From the description, the curve coming from the left seems to pass through (−3,1)(-3, 1) and (−2,−2)(-2, -2) with an open circle at (−2,−2)(-2, -2), meaning that as x→−2−x \to -2^-, the value of f(x)f(x) approaches −2-2.

Thus, lim⁡x→−2−f(x)=−2.\lim_{{x \to -2^-}} f(x) = -2.

Part 2:

Find the limit lim⁡x→−2+f(x)\lim_{{x \to -2^+}} f(x):

  • This asks for the limit of the function f(x)f(x) as xx approaches −2-2 from the right side (positive direction).
  • There is a closed circle at (−2,0)(-2, 0), which implies that as x→−2+x \to -2^+, the value of f(x)f(x) approaches 0.

Thus, lim⁡x→−2+f(x)=0.\lim_{{x \to -2^+}} f(x) = 0.

Part 3:

Find the limit lim⁡x→−2f(x)\lim_{{x \to -2}} f(x):

  • The overall limit lim⁡x→−2f(x)\lim_{{x \to -2}} f(x) exists only if the left-hand limit lim⁡x→−2−f(x)\lim_{{x \to -2^-}} f(x) and the right-hand limit lim⁡x→−2+f(x)\lim_{{x \to -2^+}} f(x) are equal.
  • In this case, since lim⁡x→−2−f(x)=−2\lim_{{x \to -2^-}} f(x) = -2 and lim⁡x→−2+f(x)=0\lim_{{x \to -2^+}} f(x) = 0, the limits are not equal, meaning the overall limit does not exist.

Thus, lim⁡x→−2f(x) does not exist.\lim_{{x \to -2}} f(x) \text{ does not exist}.

Part 4:

Find the function value f(−2)f(-2):

  • The graph shows a closed circle at the point (−2,0)(-2, 0), meaning the function is defined at x=−2x = -2, and its value is 0.

Thus, f(−2)=0.f(-2) = 0.

Final Answers:

  1. lim⁡x→−2−f(x)=−2\lim_{{x \to -2^-}} f(x) = -2
  2. lim⁡x→−2+f(x)=0\lim_{{x \to -2^+}} f(x) = 0
  3. lim⁡x→−2f(x)=does not exist\lim_{{x \to -2}} f(x) = \text{does not exist}
  4. f(−2)=0f(-2) = 0

Would you like further details or have any questions?

Here are 5 related questions to consider:

  1. What does it mean if the left and right limits at a point are not equal?
  2. How do you interpret open and closed circles on a graph in terms of limits and function values?
  3. Can a function have a defined value at a point where the limit does not exist?
  4. What is the graphical significance of a limit existing at a particular point?
  5. How do limits help in understanding the continuity of a function?

Tip: When evaluating limits, always consider the behavior of the function from both the left and right sides of the point of interest.

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Math Problem Analysis

Mathematical Concepts

Limits
Graphical Analysis
Continuity

Formulas

Limit of a function as x approaches a value

Theorems

Limit Definition
Left-Hand and Right-Hand Limits
Existence of Limits

Suitable Grade Level

Grades 10-12

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