Question

The limit \( \lim_{x \to 10} \frac{x - 10}{\sqrt{x + 6} - 4} \) is equal to:

• A. 4
• B. 8
• C. 10
• D. 16
• E. 12

Hint:

For limits involving indeterminate forms, multiplying by the conjugate is a common method to simplify the expression and resolve the limit.

Answer 1

The Correct Option is B

The expression is an indeterminate form \( \frac{0}{0} \) when \( x = 10 \). 
To resolve this, multiply both the numerator and denominator by the conjugate of the denominator: \[ \lim_{x \to 10} \frac{x - 10}{\sqrt{x + 6} - 4} \cdot \frac{\sqrt{x + 6} + 4}{\sqrt{x + 6} + 4} \] This simplifies to: \[ \lim_{x \to 10} \frac{(x - 10)(\sqrt{x + 6} + 4)}{(\sqrt{x + 6})^2 - 4^2} = \lim_{x \to 10} \frac{(x - 10)(\sqrt{x + 6} + 4)}{x + 6 - 16} \] \[ = \lim_{x \to 10} \frac{(x - 10)(\sqrt{x + 6} + 4)}{x - 10} \] Cancel \( x - 10 \) from the numerator and denominator: \[ = \lim_{x \to 10} (\sqrt{x + 6} + 4) \] Substitute \( x = 10 \): \[ = \sqrt{10 + 6} + 4 = \sqrt{16} + 4 = 4 + 4 = 8 \] Thus, the value of the limit is 8.