Questions: Find the limit. lim as x approaches -5 of (4 - sqrt(x^2 - 9)) / (x + 5) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. lim as x approaches -5 of (4 - sqrt(x^2 - 9)) / (x + 5) = (Type an integer or a simplified fraction.) B. The limit does not exist.

Solution Steps

We need to evaluate the limit: \[ \lim _{x \rightarrow -5} \frac{4 - \sqrt{x^{2} - 9}}{x + 5} \]

Substituting \( x = -5 \) directly into the expression results in an indeterminate form \( \frac{0}{0} \). Therefore, we need to simplify the expression.

To eliminate the square root, we multiply the numerator and the denominator by the conjugate of the numerator: \[ \frac{(4 - \sqrt{x^{2} - 9})(4 + \sqrt{x^{2} - 9})}{(x + 5)(4 + \sqrt{x^{2} - 9})} \] This simplifies the numerator to: \[ (4^{2} - (x^{2} - 9)) = 16 - (x^{2} - 9) = 25 - x^{2} \]

The expression now becomes: \[ \frac{25 - x^{2}}{(x + 5)(4 + \sqrt{x^{2} - 9})} \] We can factor the numerator as: \[ 25 - x^{2} = (5 - x)(5 + x) \]

Substituting this back into the limit gives: \[ \frac{(5 - x)(5 + x)}{(x + 5)(4 + \sqrt{x^{2} - 9})} \] Notice that \( 5 + x = -(x + 5) \), allowing us to cancel \( (x + 5) \): \[ \frac{-(5 - x)}{4 + \sqrt{x^{2} - 9}} \]

Now we can evaluate the limit as \( x \) approaches \(-5\): \[ \lim_{x \rightarrow -5} \frac{-(5 - x)}{4 + \sqrt{x^{2} - 9}} = \frac{-(-5 - 5)}{4 + \sqrt{(-5)^{2} - 9}} = \frac{10}{4 + \sqrt{16}} = \frac{10}{4 + 4} = \frac{10}{8} = \frac{5}{4} \]

Final Answer