KALI SEKAWAN
\(\displaystyle \lim_{x \rightarrow a} \dfrac {1}{\sqrt{x + m} - \sqrt{x + n}} \times \dfrac {\sqrt{x + m} + \sqrt{x + n}}{\sqrt{x + m} + \sqrt{x + n}} = \dfrac {\sqrt{x + m} + \sqrt{x + n}}{(x + m) - (x + n)}\)
Penjelasan
\((a + b)(a - b) = a^2 - b^2\)
\((\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = (\sqrt{a})^2 - (\sqrt{b})^2 = a - b\)
\((\sqrt{x + m} + \sqrt{x + n})(\sqrt{x + m} - \sqrt{x + n}) = (\sqrt{x + m})^2 - (\sqrt{x + n})^2 = (x + m) - (x + n)\)
Contoh
\begin{equation*}
\begin{split}
& \lim_{x \rightarrow 0} \: \frac{2 - \sqrt{4 - x}}{x} \\\\
& \lim_{x \rightarrow 0} \: \frac{2 - \sqrt{4 - x}}{x} {\color {blue} \times \frac {2 + \sqrt{4 - x}}{2 + \sqrt{4 - x}}}\\\\
& \lim_{x \rightarrow 0} \: \frac{4 - (4 - x)}{x (2 + \sqrt{4 - x})} \\\\
& \lim_{x \rightarrow 0} \: \frac{4 - 4 + x}{x (2 + \sqrt{4 - x})} \\\\
& \lim_{x \rightarrow 0} \: \frac{\cancel{x}}{\cancel{x} (2 + \sqrt{4 - x})} \\\\
& \lim_{x \rightarrow 0} \: \frac{1}{2 + \sqrt{4 - x}} \\\\
& \frac{1}{2 + \sqrt{4 - 0}} \\\\
& \frac {1}{2 + 2} \\\\
& \bbox[5px, border: 2px solid magenta] {\frac 14}
\end{split}
\end{equation*}