lim_{-3}=(6x+9)/(x^{4}+6x^{3}+9x^{2})

The limit of (6x+9)/x^4+6x^3+9x^2 as x approaches -3 does not exist. If you factor the rational expression you get:

(6x+9)/[x^4+6x^3+9x^2] = 3(2x+3) / [ (x^2)(x+3)(x+3) ]

If we let x go to -3 (from both sides) the numerator is well behaved and approaches -9, however the denominator goes to zero (but is always positive, approaching from either the left or the right). So the entire rational expression will approach negative infinity
as x goes to -3. Since there is no specific value for the limit, we say that the limit dos not exist.

## Comments

l'Hôpital's rule also called Bernoulli's rule, uses derivatives to help evaluate limits involving indeterminate forms: and you can onely use it if lim f =lim g = 0 or +–8 and other conditions. However, this condition is not satisfied in this situation so you cannot use this rule. Sorry