Math Problem Statement
Solution
Let's evaluate the limit:
Step 1: Analyze the function
The function is defined for , because the expression inside the square root must be non-negative. Therefore, the function is only valid for values of where .
Step 2: Approach the limit from the right (i.e., )
As approaches 2 from the right, the value inside the square root approaches 0 from the positive side. Therefore, the square root of a small positive number tends to 0.
Thus:
Step 3: Consider approaching the limit from the left (i.e., )
For , the expression becomes negative, which means is not defined for real numbers. Thus, the limit does not exist when approaching 2 from the left.
Final Conclusion:
The limit does not exist because the function is not defined for . However, if we restrict our attention to the right-hand limit:
Would you like more details or have any questions?
Here are 8 related questions you might consider:
- How do you evaluate limits involving absolute value functions?
- What is the difference between one-sided and two-sided limits?
- How do you evaluate the limit ?
- How does the limit of a square root function behave near a discontinuity?
- What happens to the limit ?
- How do you evaluate ?
- How do you apply the squeeze theorem to limits involving square roots?
- How do you find limits using L'Hopital's rule?
Tip: When dealing with square root functions, always check the domain of the function to ensure it is defined at the point you're approaching.
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Math Problem Analysis
Mathematical Concepts
Limits
Square Root Function
One-sided Limits
Formulas
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Theorems
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Suitable Grade Level
Advanced High School