Question

How do you find the limit of `(x^2+x-12)/(x-2)` as `x->2`?

Answer 1

`lim_(xrarr2) (x^2+x-12)/(x-2) = color(green)(+-oo)`

Explanation:

At `x=2`
`color(white)("XXX")`the numerator becomes `2^2+2-12= -6`
and
`color(white)("XXX")`the denominator becomes `2-2=0`

L'Hospital's (or L'Hopital's) Rule does not apply. L'Hopital's Rule only applies if both the numerator and denominator are `0`.

The limit of a non-negative numerator by a denominator that is approaching `0` is `+-oo` (depending upon which side the denominator is approaching "0" from).

Here is a graph of the `(x^2+x-12)/(x-2)`.
Note that as `xrarr2` the limit does not approach 5 (as implied by those attempting to use L'Hopital's Rule).