Math Problem Statement

Suppose f(x) has the following values in a table. Determine the left-hand limit \(\lim\limits_{x \to 3^-} f(x)\), the right-hand limit \(\lim\limits_{x \to 3^+} f(x)\), and the two-sided limit \(\lim\limits_{x \to 3} f(x)\) from the table.

Solution

Let's analyze the content of the image to solve the problem.

The image contains a table with values of xx and corresponding f(x)f(x), followed by three multiple-choice questions asking about the limit of f(x)f(x) as xx approaches 3 from different directions:

Table Data:

  • x=2.9x = 2.9, f(x)=6.2f(x) = 6.2
  • x=2.99x = 2.99, f(x)=6.03f(x) = 6.03
  • x=2.999x = 2.999, f(x)=6.003f(x) = 6.003
  • x=3.001x = 3.001, f(x)=5.997f(x) = 5.997
  • x=3.01x = 3.01, f(x)=5.97f(x) = 5.97
  • x=3.1x = 3.1, f(x)=5.7f(x) = 5.7

Questions:

  1. limx3f(x)\lim\limits_{x \to 3^-} f(x) (Left-hand limit)
    • Options: 66, 77, 99, Undefined.
  2. limx3+f(x)\lim\limits_{x \to 3^+} f(x) (Right-hand limit)
    • Options: 66, 77, 99, Undefined.
  3. limx3f(x)\lim\limits_{x \to 3} f(x) (Two-sided limit)
    • Options: 66, 77, 99, Undefined.

Solving the Limits:

  1. Left-hand limit limx3f(x)\lim\limits_{x \to 3^-} f(x):

    • As xx approaches 3 from the left (x=2.999x = 2.999, x=2.99x = 2.99, x=2.9x = 2.9), f(x)f(x) values are getting closer to 66.
    • Answer: 66
  2. Right-hand limit limx3+f(x)\lim\limits_{x \to 3^+} f(x):

    • As xx approaches 3 from the right (x=3.001x = 3.001, x=3.01x = 3.01, x=3.1x = 3.1), f(x)f(x) values are getting closer to 66 as well.
    • Answer: 66
  3. Two-sided limit limx3f(x)\lim\limits_{x \to 3} f(x):

    • Since both the left-hand limit and right-hand limit as xx approaches 3 are equal to 66, the two-sided limit exists and is 66.
    • Answer: 66

Summary of Answers:

  1. limx3f(x)\lim\limits_{x \to 3^-} f(x) = 6
  2. limx3+f(x)\lim\limits_{x \to 3^+} f(x) = 6
  3. limx3f(x)\lim\limits_{x \to 3} f(x) = 6

Would you like any further details or have any questions?

Related Questions:

  1. What is the significance of the left-hand and right-hand limits in determining the overall limit?
  2. How would the answers change if the left-hand and right-hand limits were different?
  3. What is the relationship between continuity and limits?
  4. Can a function have different left-hand and right-hand limits? If so, what does it imply?
  5. How does the concept of limits apply to real-world scenarios?

Tip:

When evaluating limits from a table, always observe how the function's value behaves as it approaches the point of interest from both sides to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Limits
Functions
Table of Values

Formulas

Limit from the left \(\lim\limits_{x \to c^-} f(x)\)
Limit from the right \(\lim\limits_{x \to c^+} f(x)\)
Two-sided limit \(\lim\limits_{x \to c} f(x)\)

Theorems

Limit definition
Existence of limit

Suitable Grade Level

Grades 10-12