Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 6 }$.
A) $\frac { 1 } { 6 \sqrt { x - 6 } }$
B) $\frac { 1 } { 2 \sqrt { x - 6 } }$
C) $\frac { \sqrt { x - 6 } } { 2 }$
D) $\sqrt { x - 6 }$
E) limit does not exist

Question

Quizplus · Accepted Answer

\begin{align*}
\lim _ { h ightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h } &= \lim _ { h ightarrow 0 } \frac { \sqrt { x + h - 6 } - \sqrt { x - 6 } } { h } \
&= \lim_{h ightarrow 0} \frac{(\sqrt{x+h-6} - \sqrt{x-6})(\sqrt{x+h-6} + \sqrt{x-6})}{h(\sqrt{x+h-6} + \sqrt{x-6})} \
&= \lim_{h ightarrow 0} \frac{(x+h-6) - (x-6)}{h(\sqrt{x+h-6} + \sqrt{x-6}) (\sqrt{x+h-6} + \sqrt{x-6})} \
&= \lim_{h ightarrow 0} \frac{h}{h(\sqrt{x+h-6} + \sqrt{x-6}) (\sqrt{x+h-6} + \sqrt{x-6})} \
&= \frac{1}{2(\sqrt{x-6})^2} \
&= \frac{1}{2\sqrt{x-6}} \
&= \boxed{	extbf{(B)}~\frac{1}{2\sqrt{x-6}}}.
\end{align*}

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Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$