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Question
\[\lim_{x \to 0} \frac{\sqrt{1 + x} - 1}{x}\]
Solution
\[\lim_{x \to 0} \left[ \frac{\sqrt{1 + x} - 1}{x} \right]\] It is of the form \[\frac{0}{0}\]
Rationalising the numerator:
\[\lim_{x \to 0} \left[ \frac{\left( \sqrt{1 + x} - 1 \right)\left( \sqrt{1 + x} + 1 \right)}{x\left( \sqrt{1 + x} + 1 \right)} \right]\]
= \[\lim_{x \to 0} \left[ \frac{1 + x - 1}{x\left( \sqrt{1 + x} + 1 \right)} \right]\]
= \[\frac{1}{\sqrt{1 + 0} + 1}\]
= \[\frac{1}{2}\]
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