To evaluate the limit [tex]\(\lim_{x \rightarrow 4} \frac{\sqrt{x} - \sqrt{4}}{x - 4}\)[/tex], let's proceed through the solution step-by-step.
First, we notice that directly substituting [tex]\(x = 4\)[/tex] into the expression results in an indeterminate form [tex]\(\frac{0}{0}\)[/tex]. To resolve this, we can manipulate the expression to simplify the limit:
1. Recognize that [tex]\(\sqrt{4} = 2\)[/tex]. The original limit is:
[tex]\[
\lim_{x \rightarrow 4} \frac{\sqrt{x} - 2}{x - 4}
\][/tex]
2. To simplify this expression, multiply the numerator and the denominator by the conjugate of the numerator. The conjugate of [tex]\(\sqrt{x} - 2\)[/tex] is [tex]\(\sqrt{x} + 2\)[/tex]. Thus, we have:
[tex]\[
\frac{\sqrt{x} - 2}{x - 4} \cdot \frac{\sqrt{x} + 2}{\sqrt{x} + 2} = \frac{(\sqrt{x} - 2)(\sqrt{x} + 2)}{(x - 4)(\sqrt{x} + 2)}
\][/tex]
3. Use the property that [tex]\((a - b)(a + b) = a^2 - b^2\)[/tex] to simplify the numerator:
[tex]\[
(\sqrt{x} - 2)(\sqrt{x} + 2) = x - 4
\][/tex]
Therefore, our expression becomes:
[tex]\[
\frac{x - 4}{(x - 4)(\sqrt{x} + 2)}
\][/tex]
4. We can now cancel out [tex]\(x - 4\)[/tex] in the numerator and the denominator, as long as [tex]\(x \neq 4\)[/tex]:
[tex]\[
\frac{1}{\sqrt{x} + 2}
\][/tex]
5. Now we can directly substitute [tex]\(x = 4\)[/tex] into the simplified expression:
[tex]\[
\frac{1}{\sqrt{4} + 2} = \frac{1}{2 + 2} = \frac{1}{4}
\][/tex]
Hence, the limit is:
[tex]\[
\lim_{x \rightarrow 4} \frac{\sqrt{x} - \sqrt{4}}{x - 4} = \frac{1}{4}
\][/tex]
