Question

The value of \(\lim_{x \to \infty} \sqrt{\frac{3x - 5}{x - 2}}\) is:

• A. \(\sqrt{5/2}\)
• B. \(\sqrt{3}\)
• C. Does not exist
• D. 0

Hint:

Exam Tip:
For limits at infinity of rational functions:

• Divide by the highest power of \(x\) in the denominator.
• The limit is the ratio of the leading coefficients.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We need to evaluate a limit as \(x \to \infty\). We can divide numerator and denominator by the highest power of \(x\).

Step 2: Key Formula or Approach:
\[ \lim_{x \to \infty} \frac{3x - 5}{x - 2} = \lim_{x \to \infty} \frac{3 - \frac{5}{x}}{1 - \frac{2}{x}} = 3 \] Then the square root gives \(\sqrt{3}\).

Step 3: Detailed Explanation:
\[ \lim_{x \to \infty} \sqrt{\frac{3x - 5}{x - 2}} = \sqrt{\lim_{x \to \infty} \frac{3x - 5}{x - 2}} = \sqrt{3} \]

Step 4: Final Answer:
Therefore, option (B) is correct.