Certainly! Let's find the limit of the function as [tex]\( x \)[/tex] approaches 9:
[tex]\[
\lim _{x \rightarrow 9} \frac{\sqrt{x}-3}{x-9}
\][/tex]
### Step-by-Step Solution:
1. Identify the indeterminate form:
Let's substitute [tex]\( x = 9 \)[/tex] directly into the function:
[tex]\[
\frac{\sqrt{9}-3}{9-9} = \frac{3-3}{0} = \frac{0}{0}
\][/tex]
This is an indeterminate form, so we need to simplify it further.
2. Simplify the expression:
The goal now is to try to eliminate the indeterminate form by manipulating the expression. One common technique is to multiply by the conjugate. The conjugate of [tex]\(\sqrt{x}-3\)[/tex] is [tex]\(\sqrt{x}+3\)[/tex]:
[tex]\[
\frac{\sqrt{x}-3}{x-9} \times \frac{\sqrt{x}+3}{\sqrt{x}+3}
\][/tex]
This gives us:
[tex]\[
\frac{(\sqrt{x}-3)(\sqrt{x}+3)}{(x-9)(\sqrt{x}+3)} = \frac{x - 9}{(x - 9)(\sqrt{x}+3)}
\][/tex]
3. Cancel out common terms:
We can cancel [tex]\((x - 9)\)[/tex] from the numerator and denominator, since for [tex]\( x \neq 9 \)[/tex]:
[tex]\[
\frac{x - 9}{(x - 9)(\sqrt{x}+3)} = \frac{1}{\sqrt{x}+3}
\][/tex]
4. Evaluate the limit:
Now we substitute [tex]\( x = 9 \)[/tex] into the simplified expression:
[tex]\[
\lim_{x \rightarrow 9} \frac{1}{\sqrt{x}+3} = \frac{1}{\sqrt{9}+3} = \frac{1}{3+3} = \frac{1}{6}
\][/tex]
Hence, the limit is:
[tex]\[
\lim _{x \rightarrow 9} \frac{\sqrt{x}-3}{x-9} = \frac{1}{6}
\][/tex]
