Math Problem Statement
Find lim_{x→2} f(x) using the provided graph.
Solution
The problem asks for using the graph.
To solve this, let's examine the graph at :
- Left-hand limit: As approaches 2 from the left (values slightly less than 2), the function appears to approach the value . This is indicated by the curve coming downwards toward the point where from the left.
- Right-hand limit: As approaches 2 from the right (values slightly greater than 2), the function also approaches the value . The curve on the right side of 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,
Would you like more details or have any questions?
Here are 5 related questions to expand on this concept:
- What is a one-sided limit, and how does it relate to two-sided limits?
- How can we determine if a function is continuous at a point using limits?
- What happens if the left-hand and right-hand limits are not equal?
- How do discontinuities affect limits?
- 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)