$\text { Find } \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \text { for } f(x)=\sqrt{x-7} \text {. }$

A) $ \frac{1}{7 \sqrt{x-7}} $
B) $ \frac{1}{2 \sqrt{x-7}} $
C) $ \frac{\sqrt{x-7}}{2} $
D) $ \sqrt{x-7} $
E) limit does not exist

Question

$\text { Find } \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \text { for } f(x)=\sqrt{x-7} \text {. }$

A) $ \frac{1}{7 \sqrt{x-7}} $
B) $ \frac{1}{2 \sqrt{x-7}} $
C) $ \frac{\sqrt{x-7}}{2} $
D) $ \sqrt{x-7} $
E) limit does not exist

Quizplus · Accepted Answer

First, we need to apply the limit definition of the derivative:

$$f'(x)=\lim_{h	o 0}\frac{f(x+h)-f(x)}{h}$$

Applying it to our function $f(x)=\sqrt{x-7}$, we get:

\begin{align*}
f'(x)&=\lim_{h	o 0}\frac{f(x+h)-f(x)}{h}\
&=\lim_{h	o 0}\frac{\sqrt{x+h-7}-\sqrt{x-7}}{h}\
&=\lim_{h	o 0}\frac{\sqrt{x+h-7}-\sqrt{x-7}}{h}\cdot\frac{\sqrt{x+h-7}+\sqrt{x-7}}{\sqrt{x+h-7}+\sqrt{x-7}}\
&=\lim_{h	o 0}\frac{(x+h-7)-(x-7)}{h(\sqrt{x+h-7}+\sqrt{x-7})}\
&=\lim_{h	o 0}\frac{h}{h(\sqrt{x+h-7}+\sqrt{x-7})}\
&=\lim_{h	o 0}\frac{1}{\sqrt{x+h-7}+\sqrt{x-7}}\
&=\frac{1}{2\sqrt{x-7}}\qquad(	ext{since }h	o 0\implies\sqrt{x+h-7}	o\sqrt{x-7})
\end{align*}

Therefore, the answer is B.

Find lim⁡h→0f(x+h)−f(x)h for f(x)=x−7. \text { Find } \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \text { for } f(x)=\sqrt{x-7} \text {. } Find limh→0​hf(x+h)−f(x)​ for f(x)=x−7​.

$\text { Find } \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \text { for } f(x)=\sqrt{x-7} \text {. }$