To evaluate the limit [tex]\(\lim_{x \rightarrow 9} \frac{x - 9}{\sqrt{x} - 3}\)[/tex], follow these steps:
1. Direct Substitution: First, we try to substitute [tex]\(x = 9\)[/tex] directly into the expression.
[tex]\[
\frac{9 - 9}{\sqrt{9} - 3} = \frac{0}{0}
\][/tex]
This results in an indeterminate form ([tex]\(\frac{0}{0}\)[/tex]), indicating that we need to simplify or manipulate the expression further to find the limit.
2. Simplifying the Expression:
To simplify, let's try to factor and reduce the expression. Rewrite the numerator [tex]\(x - 9\)[/tex] in a form that can potentially cancel the denominator [tex]\(\sqrt{x} - 3\)[/tex].
Notice that [tex]\(x = (\sqrt{x})^2\)[/tex], so:
[tex]\[
x - 9 = (\sqrt{x})^2 - 9 = (\sqrt{x} - 3)(\sqrt{x} + 3)
\][/tex]
This is the difference of squares. Now, substitute this back into the original limit expression:
[tex]\[
\frac{x - 9}{\sqrt{x} - 3} = \frac{(\sqrt{x} - 3)(\sqrt{x} + 3)}{\sqrt{x} - 3}
\][/tex]
3. Canceling the Common Factor:
We can now cancel out the [tex]\(\sqrt{x} - 3\)[/tex] term in the numerator and denominator:
[tex]\[
\frac{(\sqrt{x} - 3)(\sqrt{x} + 3)}{\sqrt{x} - 3} = \sqrt{x} + 3 \quad \text{for} \quad x \neq 9
\][/tex]
4. Evaluate the Limit:
Now that we have simplified the expression, we can directly substitute [tex]\(x = 9\)[/tex] into [tex]\(\sqrt{x} + 3\)[/tex]:
[tex]\[
\lim_{x \rightarrow 9} (\sqrt{x} + 3) = \sqrt{9} + 3 = 3 + 3 = 6
\][/tex]
Hence, the limit is:
[tex]\[
\lim_{x \rightarrow 9} \frac{x - 9}{\sqrt{x} - 3} = 6
\][/tex]
