Answer to Question #300981 in Algebra for golden

Question

(69) Why is it impossible to find the instantaneous rate of change of f(x)= x^2-9/x-3 at x=3? THE SLASH IS THE FRACTION SYMBOL.

Answer 1

Explanations & Calculations

• It is because at $\small x =3$ the graph has a vertical asymptote.
• So the graph approaches infinity as it reaches $\small x \to 3$.
• The rate of change behaves the same way $\small$$\small f'(x)_{x=3} = \frac{df(x)}{dx}_{x=3}\to -\infty$
• The rate of change is given by

$\qquad\qquad \begin{aligned} \small f'(x) &=\small 2x-\frac{9}{(x-3)^2} \end{aligned}$