Question

\(\lim_{x \to 11} \frac{x - 11}{\sqrt{x^2 + 48} - 13} =\)

• A. \(\frac{11}{13}\)
• B. \(\frac{13}{11}\)
• C. \(\frac{26}{11}\)
• D. \(\frac{11}{26}\)
• E. 0

Hint:

When you see \(\sqrt{\cdots} - a\), multiply numerator and denominator by the conjugate \(\sqrt{\cdots} + a\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Rationalize the denominator to handle the indeterminate form \(\frac{0}{0}\).

Step 2: Detailed Explanation:
\[ \lim_{x \to 11} \frac{x - 11}{\sqrt{x^2 + 48} - 13} = \lim_{x \to 11} \frac{(x - 11)(\sqrt{x^2 + 48} + 13)}{(x^2 + 48) - 169} \] \[ = \lim_{x \to 11} \frac{(x - 11)(\sqrt{x^2 + 48} + 13)}{x^2 - 121} = \lim_{x \to 11} \frac{(x - 11)(\sqrt{x^2 + 48} + 13)}{(x - 11)(x + 11)} \] \[ = \lim_{x \to 11} \frac{\sqrt{x^2 + 48} + 13}{x + 11} = \frac{\sqrt{121 + 48} + 13}{22} = \frac{\sqrt{169} + 13}{22} = \frac{13 + 13}{22} = \frac{26}{22} = \frac{13}{11} \]

Step 3: Final Answer:
The limit is \(\frac{13}{11}\).