Math Problem Statement

limx2x2\lim_{\text{x}\rarr2}\sqrt{x-2}

Solution

Let's evaluate the limit:

limx2x2\lim_{x \to 2} \sqrt{x - 2}

Step 1: Analyze the function

The function x2\sqrt{x - 2} is defined for x2x \geq 2, because the expression inside the square root x2x - 2 must be non-negative. Therefore, the function is only valid for values of xx where x2x \geq 2.

Step 2: Approach the limit from the right (i.e., x2+x \to 2^+)

As xx approaches 2 from the right, the value inside the square root x2x - 2 approaches 0 from the positive side. Therefore, the square root of a small positive number tends to 0.

Thus: limx2+x2=0\lim_{x \to 2^+} \sqrt{x - 2} = 0

Step 3: Consider approaching the limit from the left (i.e., x2x \to 2^-)

For x<2x < 2, the expression x2x - 2 becomes negative, which means x2\sqrt{x - 2} is not defined for real numbers. Thus, the limit does not exist when approaching 2 from the left.

Final Conclusion:

The limit limx2x2\lim_{x \to 2} \sqrt{x - 2} does not exist because the function is not defined for x<2x < 2. However, if we restrict our attention to the right-hand limit:

limx2+x2=0\lim_{x \to 2^+} \sqrt{x - 2} = 0

Would you like more details or have any questions?

Here are 8 related questions you might consider:

  1. How do you evaluate limits involving absolute value functions?
  2. What is the difference between one-sided and two-sided limits?
  3. How do you evaluate the limit limx01x\lim_{x \to 0} \frac{1}{x}?
  4. How does the limit of a square root function behave near a discontinuity?
  5. What happens to the limit limx1x2\lim_{x \to \infty} \frac{1}{x^2}?
  6. How do you evaluate limx0x\lim_{x \to 0} \sqrt{x}?
  7. How do you apply the squeeze theorem to limits involving square roots?
  8. 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