f(x) = \frac{1}{x - 3}
Use the limit definition of the derivative on page 156 to find
(i) f'( -1 ) =
(ii) f'( 1 ) =
(iii) f'( 5 ) =
(iv) f'( 7 ) =

To avoid calculating four separate limits, I suggest that you evaluate the derivative at the point when x = a. Once you have the derivative, you can just plug in those four values for "a" to get the answers.

Question

f(x) = \frac{1}{x - 3}
Use the limit definition of the derivative on page 156 to find
(i) f'( -1 ) =
(ii) f'( 1 ) =
(iii) f'( 5 ) =
(iv) f'( 7 ) =

To avoid calculating four separate limits, I suggest that you evaluate the derivative at the point when x = a. Once you have the derivative, you can just plug in those four values for "a" to get the answers.

anonymous · Answer

$$f(a)=\frac{1}{a-3}~~\Rightarrow~~f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
$$\begin{align*}f'(a)&=\lim_{h\to0}\frac{\dfrac{1}{a+h-3}-\dfrac{1}{a-3}}{h}\\
&=\lim_{h\to0}\frac{\dfrac{a-3}{(a+h-3)(a-3)}-\dfrac{a+h-3}{(a-3)(a+h-3)}}{h}\\
&=\lim_{h\to0}\frac{\dfrac{-h}{(a+h-3)(a-3)}}{h}\\
&=\lim_{h\to0}\frac{-1}{(a+h-3)(a-3)}\\\
&=\cdots
\end{align*}$$

anonymous · Answer

So you'll get some function of $a$. To find (a)-(d), all you have to do is plug in the given $a$.

anonymous · Answer

Sorry, I mean (i)-(iv)...

anonymous · Answer

thank you so much! will my answers include "h" in them or no?

anonymous · Answer

never mind i got it now thank you so much again!!

anonymous · Answer

You're welcome!