Question

find the following limit or state that it does not exist. (limlimits_{x \to 4} \frac{sqrt{x} - 2}{x - 4}) simplify the given limit. (limlimits_{x \to 4} \frac{sqrt{x} - 2}{x - 4} = limlimits_{x \to 4} (square)) (simplify your answer.)

Explanation:

Step1: Factor the denominator

Notice that \(x - 4\) is a difference of squares, so \(x - 4=(\sqrt{x})^2 - 2^2 = (\sqrt{x}-2)(\sqrt{x}+2)\).
The limit becomes \(\lim_{x
ightarrow4}\frac{\sqrt{x}-2}{(\sqrt{x}-2)(\sqrt{x}+2)}\).

Step2: Cancel common factors

Cancel the common factor \(\sqrt{x}-2\) (assuming \(\sqrt{x}-2
eq0\), which is true as \(x
ightarrow4\) but \(x
eq4\) in the limit process).
We get \(\lim_{x
ightarrow4}\frac{1}{\sqrt{x}+2}\).

Answer:

\(\frac{1}{\sqrt{x}+2}\)