evaluate: lim h tends to 0 (under root(x+h)-under root (x))/ h

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Solution

$\lim_{h \to 0} \frac{\sqrt{x + h} - \sqrt{x}}{h} So \lim_{h \to 0} \frac{\sqrt{x + h} - \sqrt{x}}{h} \times \frac{\sqrt{x + h} + \sqrt{x}}{\sqrt{x + h} + \sqrt{x}} = \lim_{h \to 0} \frac{x + h - x}{h \times (\sqrt{x + h} + \sqrt{x})} = \lim_{h \to 0} \frac{1}{\sqrt{x + h} + \sqrt{x}} (put h = 0, as indeterminate form is removed) = \frac{1}{2 \sqrt{x}} (ans)$

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