Given g(x)=5/(x−3), evaluate and simplify: (g(5+h)−g(5))/h =?

Mar 16, 2018

$- \frac{5}{2 h + 4}$

Explanation:

We have: $g \left(x\right) = \frac{5}{x - 3}$

In order to evaluate $\frac{g \left(5 + h\right) - g \left(5\right)}{h}$, we simply substitute $5 + h$ and $5$ in place of $x$ in $g \left(x\right)$:

$R i g h t a r r o w \frac{g \left(5 + h\right) - g \left(5\right)}{h} = \frac{\frac{5}{\left(5 + h\right) - 3} - \frac{5}{\left(5\right) - 3}}{h}$

$R i g h t a r r o w \frac{g \left(5 + h\right) - g \left(5\right)}{h} = \frac{\frac{5}{h + 2} - \frac{5}{2}}{h}$

$R i g h t a r r o w \frac{g \left(5 + h\right) - g \left(5\right)}{h} = \frac{\frac{5 \cdot 2 - 5 \cdot \left(h + 2\right)}{2 \cdot \left(h + 2\right)}}{h}$

$R i g h t a r r o w \frac{g \left(5 + h\right) - g \left(5\right)}{h} = \frac{\frac{10 - 5 h - 10}{2 \cdot \left(h + 2\right)}}{h}$

$R i g h t a r r o w \frac{g \left(5 + h\right) - g \left(5\right)}{h} = \frac{- 5 h}{2 \cdot \left(h + 2\right)} \cdot \frac{1}{h}$

$\therefore \frac{g \left(5 + h\right) - g \left(5\right)}{h} = - \frac{5}{2 h + 4}$

Mar 16, 2018

$- \frac{5}{2 \left(h + 2\right)}$

Explanation:

$\text{evaluating each term separately}$

$g \left(5 + h\right) = \frac{5}{5 + h - 3} = \frac{5}{2 + h}$

$f \left(5\right) = \frac{5}{5 - 3} = \frac{5}{2}$

$\Rightarrow \left(g \left(5 + h\right) - g \left(5\right)\right)$

$= \frac{5}{2 + h} - \frac{5}{2}$

$= \frac{10 - 5 \left(2 + h\right)}{2 \left(2 + h\right)} = \frac{- 5 h}{2 \left(2 + h\right)}$

$\Rightarrow \frac{g \left(5 + h\right) - g \left(5\right)}{h}$

$= \frac{- 5 \cancel{h}}{\cancel{h} 2 \left(2 + h\right)} = - \frac{5}{2 \left(2 + h\right)}$