Question

current objective calculate the derivative of a given function at a point using a limit
question
given (f(x)=\frac{3}{x - 2}), find (f(4)) using the definition of a derivative.
provide your answer below:
(f(4)=square)

Explanation:

Step1: Recall derivative definition

The definition of the derivative of a function $y = f(x)$ at a point $x = a$ is $f^{\prime}(a)=\lim_{h
ightarrow0}\frac{f(a + h)-f(a)}{h}$. Here, $a = 4$ and $f(x)=\frac{3}{x - 2}$. So $f(4 + h)=\frac{3}{(4 + h)-2}=\frac{3}{h + 2}$ and $f(4)=\frac{3}{4-2}=\frac{3}{2}$.

Step2: Substitute into the formula

$f^{\prime}(4)=\lim_{h
ightarrow0}\frac{\frac{3}{h + 2}-\frac{3}{2}}{h}=\lim_{h
ightarrow0}\frac{\frac{3\times2-3(h + 2)}{2(h + 2)}}{h}=\lim_{h
ightarrow0}\frac{6-3h - 6}{2h(h + 2)}$.

Step3: Simplify the numerator

The numerator $6-3h - 6=-3h$. So the limit becomes $\lim_{h
ightarrow0}\frac{-3h}{2h(h + 2)}$.

Step4: Cancel out the common - factor

Since $h
eq0$ (as we are taking the limit as $h$ approaches 0, not setting $h = 0$), we can cancel out the $h$ terms. We get $\lim_{h
ightarrow0}\frac{-3}{2(h + 2)}$.

Step5: Evaluate the limit

Substitute $h = 0$ into $\frac{-3}{2(h + 2)}$. We have $\frac{-3}{2(0 + 2)}=-\frac{3}{4}$.

Answer:

$-\frac{3}{4}$