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 does not exist.

Andrew G.

03/26/15