Math Problem Statement

For a function f that is continuous at x = 2, with lim f(x) = 4 as x approaches 2 from the left, and lim f(x) = 4 as x approaches 2 from the right, find f(2).

Solution

In this problem, we are given that the function ff is continuous at x=2x = 2, and we know the following:

  • limx2f(x)=4\lim\limits_{x \to 2^-} f(x) = 4 (the left-hand limit as xx approaches 2)
  • limx2+f(x)=4\lim\limits_{x \to 2^+} f(x) = 4 (the right-hand limit as xx approaches 2)

Since the function is continuous at x=2x = 2, the value of the function at x=2x = 2 must be equal to the limit of the function as xx approaches 2 from both sides. In other words:

f(2)=limx2f(x)f(2) = \lim\limits_{x \to 2} f(x)

Given that both the left-hand and right-hand limits are equal to 4, we conclude that:

f(2)=4f(2) = 4

Thus, the correct answer is:

f(2)=4f(2) = 4

Would you like further details or have any questions about this solution?

Here are some follow-up questions for deeper understanding:

  1. What does it mean for a function to be continuous at a point?
  2. What is the difference between one-sided limits and a two-sided limit?
  3. Can a function have different left-hand and right-hand limits at a point and still be continuous?
  4. How can you use limits to identify discontinuities in a function?
  5. How would you define a removable discontinuity?

Tip: A function is continuous at a point if its value at that point equals the limit as you approach the point from both directions.

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Math Problem Analysis

Mathematical Concepts

Continuity
Limits
One-sided limits

Formulas

f(2) = lim(x -> 2) f(x)

Theorems

Definition of continuity
One-sided limits must be equal for continuity

Suitable Grade Level

Grades 10-12