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 2, 3) and (0, 2). The second line segment extends from the open circle at (0, 2) to a closed circle at (2, 3). The first curve begins at a closed circle at (negative 2, negative 4) and rises right to left at an increasing rate passing through an open circle at (negative 3, negative 3) and through (negative 4, 0). The second curve begins at an open circle at (2, negative 4) and rises left to right at an increasing rate passing through (4, 0). A solid circle is plotted at (negative 3, 1). Question content area bottom Part 1 ​(A) Find the limit ModifyingBelow lim With font size decreased by 1 x right arrow minus 3 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 3 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.

Solution

Let's break down the problem step by step to find the limit of f(x)f(x) as xx approaches −3-3 from the left, denoted as:

lim⁡x→−3−f(x)\lim_{{x \to -3^-}} f(x)

Part (A)

We are asked to find this limit by observing the behavior of f(x)f(x) as xx approaches −3-3 from the left side (denoted as x→−3−x \to -3^-).

From the description:

  • There is a curve that begins at a closed circle at (−2,−4)(-2, -4) and rises as it moves left toward (−3,−3)(-3, -3).
  • There is also a solid circle at (−3,1)(-3, 1).

However, we are specifically interested in the behavior of f(x)f(x) as x→−3−x \to -3^-. Based on the curve passing through (−4,0)(-4, 0) and (−3,−3)(-3, -3), as xx approaches −3-3 from the left, the function value seems to approach −3-3.

Thus, we conclude:

lim⁡x→−3−f(x)=−3\lim_{{x \to -3^-}} f(x) = -3

Do you want details or have any questions?
Here are 5 related questions for you:

  1. What is the right-hand limit lim⁡x→−3+f(x)\lim_{{x \to -3^+}} f(x)?
  2. Does the function have a discontinuity at x=−3x = -3?
  3. How does the limit change when approaching other points, like x=0x = 0?
  4. Can you identify any other points where the function is not continuous?
  5. How would you describe the overall behavior of the function graphically?

Tip: When evaluating limits graphically, always consider the direction of approach and whether the function has jumps or holes at specific points.

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Math Problem Analysis

Mathematical Concepts

Limits
Piecewise Functions
Graphical Analysis

Formulas

Limit of a function: lim(x→c) f(x)

Theorems

Limit Theorem
Continuity Theorem

Suitable Grade Level

Grades 10-12

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