# Limit on a five term polynomial

it has been two years since I have taken or used calculus, and I am having some trouble with factoring a polynomial in order to take a limit on it. I have searched for previous similar questions here, but I have been unable to find anything helpful. Here is my problem:

I have: \begin{equation} \lim_{x \to\ 1}\frac{{x^4+3x^3-13x^2-27x+36}}{x^2+3x-4} \end{equation}

So, how do I approach this? When I try long division, I end up with a remainder, e.g.:

\begin{equation} _{x \to\ 4}\frac{{x^4+3x^3-13x^2-27x+36}}{x^2+3x-4} = (x+3)(x+3)+\frac{{24x+72}}{(x+1)(x-4)} \end{equation}

Plugging in the limit to what I came up with via long division just yields an undefined result, so I'm obviously doing something wrong.

If anyone can help, I would really appreciate it!

Thanks!

• There was an arithmetical error in the long division. If done correctly we get quotient $x^2-9$ and remainder $0$. – André Nicolas Jan 31 '16 at 4:30

Since the numerator is non-zero when $x = 4$, the limit does not exist. We can write the rational function as
$$R(x) \cdot \frac{1}{x - 4}$$
where $R(4) \ne 0$ (and $R$ is continuous on a neighborhood of $4$). But since
$$\lim_{x \to 4} \frac{1}{x - 4}$$ does not exist, your limit does not exist.