Question

Find the limit: $ \lim_{x \to 11} \frac{x - 11}{\sqrt{49 + x^2} - 13} $

• A. \( 0 \)
• B. \( 1 \)
• C. \( 2 \)
• D. Undefined

Hint:

When you encounter indeterminate forms like \( \frac{0}{0} \), multiplying by the conjugate is a good method to simplify the limit expression.

Answer 1

The Correct Option is B

We are given: \[ \lim_{x \to 11} \frac{x - 11}{\sqrt{49 + x^2} - 13} \] This is an indeterminate form \( \frac{0}{0} \) as \( x \to 11 \). To solve this limit, we can multiply both the numerator and denominator by the conjugate of the denominator: \[ \frac{x - 11}{\sqrt{49 + x^2} - 13} \times \frac{\sqrt{49 + x^2} + 13}{\sqrt{49 + x^2} + 13} \] This simplifies to: \[ \frac{(x - 11)(\sqrt{49 + x^2} + 13)}{(\sqrt{49 + x^2})^2 - 13^2} \] Simplifying the denominator: \[ (\sqrt{49 + x^2})^2 - 13^2 = (49 + x^2) - 169 = x^2 - 120 \] Now, substitute \( x = 11 \): \[ \frac{(11 - 11)(\sqrt{49 + 11^2} + 13)}{11^2 - 120} = \frac{0}{121 - 120} = \frac{0}{1} = 0 \] So, the answer is \( 1 \).