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

Using the definition of the derivative, we have:

\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}}{h}\cdot\frac{\sqrt{x+h-6}+\sqrt{x-6}}{\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})} \
&= \lim_{hightarrow 0}\frac{h}{h(\sqrt{x+h-6}+\sqrt{x-6})} \
&= \lim_{hightarrow 0}\frac{1}{\sqrt{x+h-6}+\sqrt{x-6}} \
&= \frac{1}{2\sqrt{x-6}}.
\end{align*}

Therefore, the answer is (B).

Find lim⁡h→0f(x+h)−f(x)h\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }limh→0​hf(x+h)−f(x)​

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