Sure, let's solve the following limit step-by-step:
[tex]\[
\lim _{x \rightarrow 4} \frac{x-4}{\sqrt{x}-\sqrt{4}}
\][/tex]
First, we treat [tex]\(x-4\)[/tex] as a difference of squares. Notice that [tex]\(x\)[/tex] can be rewritten to show the difference of squares:
[tex]\[
x - 4 = (\sqrt{x})^2 - (2)^2
\][/tex]
Next, we apply the difference of squares factorization:
[tex]\[
(\sqrt{x})^2 - (2)^2 = (\sqrt{x} - 2)(\sqrt{x} + 2)
\][/tex]
So, the given limit becomes:
[tex]\[
\lim _{x \rightarrow 4} \frac{(\sqrt{x} - 2)(\sqrt{x} + 2)}{\sqrt{x} - \sqrt{4}}
\][/tex]
Note that [tex]\(\sqrt{4} = 2\)[/tex], so we can simplify the expression further by canceling out [tex]\(\sqrt{x} - 2\)[/tex]:
[tex]\[
\lim _{x \rightarrow 4} \frac{(\sqrt{x} - 2)(\sqrt{x} + 2)}{\sqrt{x} - 2}
\][/tex]
Cancelling [tex]\(\sqrt{x} - 2\)[/tex] from both the numerator and the denominator (as long as [tex]\(x \neq 4\)[/tex]):
[tex]\[
\lim _{x \rightarrow 4} (\sqrt{x} + 2)
\][/tex]
Now, we can evaluate the limit by substituting [tex]\(x = 4\)[/tex] in the simplified expression:
[tex]\[
\sqrt{4} + 2 = 2 + 2 = 4
\][/tex]
So, the simplified expression and the value of the limit are:
[tex]\[
\boxed{\sqrt{x} + 2}
\][/tex]
[tex]\[
\boxed{4}
\][/tex]
