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