Math Problem Statement

Find lim_{x→2} f(x) using the provided graph.

Solution

The problem asks for limx2f(x)\lim_{x \to 2} f(x) using the graph.

To solve this, let's examine the graph at x=2x = 2:

  • Left-hand limit: As xx approaches 2 from the left (values slightly less than 2), the function appears to approach the value 44. This is indicated by the curve coming downwards toward the point where x=2x = 2 from the left.
  • Right-hand limit: As xx approaches 2 from the right (values slightly greater than 2), the function also approaches the value 44. The curve on the right side of x=2x = 2 approaches the same point.

Since both the left-hand and right-hand limits exist and are equal to 4, the two-sided limit exists and is also equal to 4.

Thus, limx2f(x)=4.\lim_{x \to 2} f(x) = 4.

Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:

  1. What is a one-sided limit, and how does it relate to two-sided limits?
  2. How can we determine if a function is continuous at a point using limits?
  3. What happens if the left-hand and right-hand limits are not equal?
  4. How do discontinuities affect limits?
  5. Can a limit exist even if the function value at that point does not?

Tip: Always check both one-sided limits when determining a two-sided limit to ensure they are equal!

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Math Problem Analysis

Mathematical Concepts

Limits
Continuity
Left-hand limit
Right-hand limit

Formulas

lim_{x→c} f(x)
Left-hand limit: lim_{x→c-} f(x)
Right-hand limit: lim_{x→c+} f(x)

Theorems

Existence of two-sided limits
Limit definition: If left-hand and right-hand limits are equal, the two-sided limit exists.

Suitable Grade Level

Grades 11-12 (Pre-Calculus, Calculus)