To find the limit [tex]\(\lim_{x \to -6} \frac{\sqrt{10-x} - 4}{x + 6}\)[/tex], let's proceed carefully and analytically.
### Step 1: Identify the Indeterminate Form
Firstly, substitute [tex]\( x = -6 \)[/tex] into the expression to check the form:
[tex]\[
\frac{\sqrt{10 - (-6)} - 4}{-6 + 6} = \frac{\sqrt{16} - 4}{0} = \frac{4 - 4}{0} = \frac{0}{0}
\][/tex]
This is an indeterminate form, [tex]\(\frac{0}{0}\)[/tex], so we need to use algebraic techniques to resolve it.
### Step 2: Rationalize the Numerator
When we have a square root in the numerator, a good technique is to rationalize the numerator. To do this, we multiply the numerator and the denominator by the conjugate of the numerator:
[tex]\[
\frac{\sqrt{10 - x} - 4}{x + 6} \cdot \frac{\sqrt{10 - x} + 4}{\sqrt{10 - x} + 4}
\][/tex]
### Step 3: Simplify the Expression
Multiply out both the numerator and the denominator:
Numerator:
[tex]\[
(\sqrt{10 - x} - 4)(\sqrt{10 - x} + 4) = (\sqrt{10 - x})^2 - 4^2 = (10 - x) - 16 = -x - 6
\][/tex]
Denominator:
[tex]\[
(x + 6)(\sqrt{10 - x} + 4)
\][/tex]
So the expression becomes:
[tex]\[
\frac{-x - 6}{(x + 6)(\sqrt{10 - x} + 4)}
\][/tex]
### Step 4: Factor and Cancel
Notice that [tex]\(-x - 6 = -(x + 6)\)[/tex], thus we can rewrite:
[tex]\[
\frac{-(x + 6)}{(x + 6)(\sqrt{10 - x} + 4)}
\][/tex]
Now, we can cancel the [tex]\(x + 6\)[/tex] term in the numerator and denominator:
[tex]\[
\frac{-1}{\sqrt{10 - x} + 4}
\][/tex]
### Step 5: Evaluate the Limit
Now substitute [tex]\( x = -6 \)[/tex] into the simplified expression:
[tex]\[
\lim_{x \to -6} \frac{-1}{\sqrt{10 - x} + 4} = \frac{-1}{\sqrt{10 - (-6)} + 4} = \frac{-1}{\sqrt{16} + 4} = \frac{-1}{4 + 4} = \frac{-1}{8}
\][/tex]
### Final Answer
So the limit is:
[tex]\[
\boxed{-\frac{1}{8}}
\][/tex]
