To solve this limit problem, we can use algebraic manipulation to simplify the expression. The expression involves a square root, so we can multiply the numerator and the denominator by the conjugate of the numerator to eliminate the square root. This will help us simplify the expression and evaluate the limit as \( x \) approaches -4.
We are given the limit expression:
\[
\lim _{x \rightarrow-4} \frac{\sqrt{x^{2}+9}-5}{x+4}
\]
To simplify the expression, multiply the numerator and the denominator by the conjugate of the numerator, \(\sqrt{x^2 + 9} + 5\):
\[
\frac{\sqrt{x^2 + 9} - 5}{x + 4} \cdot \frac{\sqrt{x^2 + 9} + 5}{\sqrt{x^2 + 9} + 5}
\]
The numerator becomes:
\[
(\sqrt{x^2 + 9} - 5)(\sqrt{x^2 + 9} + 5) = (x^2 + 9) - 25 = x^2 - 16
\]
The denominator becomes:
\[
(x + 4)(\sqrt{x^2 + 9} + 5)
\]
Thus, the expression simplifies to:
\[
\frac{x^2 - 16}{(x + 4)(\sqrt{x^2 + 9} + 5)}
\]
Notice that \(x^2 - 16\) can be factored as \((x - 4)(x + 4)\). Therefore, the expression becomes:
\[
\frac{(x - 4)(x + 4)}{(x + 4)(\sqrt{x^2 + 9} + 5)}
\]
Cancel the \((x + 4)\) terms:
\[
\frac{x - 4}{\sqrt{x^2 + 9} + 5}
\]
Now, evaluate the limit as \(x\) approaches \(-4\):
\[
\lim_{x \to -4} \frac{x - 4}{\sqrt{x^2 + 9} + 5} = \frac{-4 - 4}{\sqrt{(-4)^2 + 9} + 5} = \frac{-8}{\sqrt{16 + 9} + 5} = \frac{-8}{\sqrt{25} + 5} = \frac{-8}{5 + 5} = \frac{-8}{10} = -0.8
\]
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