Finding $\displaystyle\lim_{{{h}\to{0}}}{\frac{{{\left(\sqrt{{{9}+{h}}}-{3}\right)}}}{{{h}}}}$

Question

Finding

lim_{h \to 0} \frac{(\sqrt{9 + h} - 3)}{h}

Answer 1

$\lim_{h \to 0} \frac{\sqrt{9 + h} - 3}{h}$

$= \lim_{h \to 0} \frac{(\sqrt{9 + h} - 3) (\sqrt{9 + h} + 3)}{h (\sqrt{9 + h} + 3)}$
$= lim_{h \to 0} \frac{{(\sqrt{9 + h})}^{2} - 3^{2}}{h (\sqrt{9 + h} + 3)}$
$= lim_{h \to 0} \frac{9 + h - 9}{h (\sqrt{9 + h} + 3)}$
$= lim_{h \to 0} \frac{h}{h (\sqrt{9 + h} + 3)}$
$= lim_{h \to 0} \frac{1}{\sqrt{9 + h} + 3}$
$= \frac{1}{\sqrt{9} + 3}$
$= \frac{1}{6}$

Finding \(\displaystyle\lim_{{{h}\to{0}}}{\frac{{{\left(\sqrt{{{9}+{h}}}-{3}\right)}}}{{{h}}}}\)

Answers (1)

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