Math Problem Statement
Solution
In this problem, we are given that the function is continuous at , and we know the following:
- (the left-hand limit as approaches 2)
- (the right-hand limit as approaches 2)
Since the function is continuous at , the value of the function at must be equal to the limit of the function as approaches 2 from both sides. In other words:
Given that both the left-hand and right-hand limits are equal to 4, we conclude that:
Thus, the correct answer is:
Would you like further details or have any questions about this solution?
Here are some follow-up questions for deeper understanding:
- What does it mean for a function to be continuous at a point?
- What is the difference between one-sided limits and a two-sided limit?
- Can a function have different left-hand and right-hand limits at a point and still be continuous?
- How can you use limits to identify discontinuities in a function?
- 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