Question

\sqrt{x} " align="absmiddle" class="latex-formula">
Lim h=0
f(x+h) -f(x)/h​

Answer 1

Answer:    $\dfrac{1}{2\sqrt x}$

Step-by-step explanation:

$\lim_{h \to 0} f(x)=\dfrac{f(x+h)-f(x)}{h}$

f(x) = $\sqrt x$

f(x+h) = $\sqrt{x+h}$

$\lim_{h \to 0} f(x)=\dfrac{\sqrt{x+h}-\sqrt x}{h}$

                   $=\dfrac{\sqrt{x+h}-\sqrt x}{h}\bigg(\dfrac{\sqrt{x+h}+\sqrt x}{\sqrt{x+h}+\sqrt x}\bigg)$

                   $=\dfrac{(x + h)-(x)}{h(\sqrt{x+h}+\sqrt x)}$

                   $=\dfrac{h}{h(\sqrt{x+h}+\sqrt x)}$

                   $=\dfrac{1}{\sqrt{x+h}+\sqrt x}$

                  $=\dfrac{1}{\sqrt{x+0}+\sqrt x}$

                  $=\dfrac{1}{2\sqrt x}$