Question

How do you find `\lim _ { x \rightarrow 9} ( \frac { x ^ { 2} - 9} { x - 9} )`?

Answer 1

Not defined.

Explanation:

`Lt_(x->9)((x^2-9)/(x-9))`

Let `x=9+Deltax`, then as `x->9`, `Deltax->0`

and `Lt_(x->9)((x^2-9)/(x-9))`

= `Lt_(Deltax->0)(((9+Deltax)^2-9)/(9+Deltax-9))`

= `Lt_(Deltax->0)((81+18Deltax+Deltax^2-9)/(9+Deltax-9))`

= `Lt_(Deltax->0)((72+18Deltax+Deltax^2)/(Deltax))`

= `Lt_(Deltax->0)(72/(Deltax)+18Deltax+Deltax)`

= `oo`

But observe that when `x=9-Deltax` it will `-oo` and it is discontinuous at `x=9` and hence limit is not defined at `x=9`
graph{(x^2-9)/(x-9) [-160, 160, -80, 80]}