Finding one sided limits algebraically I was wondering what the best method was for proving this limit algebraically:
$$\lim_{x \to 1}\frac{3x^4-8x^3+5}{x^3-x^2-x+1}$$
I know the answer to this question is ;
$$\lim_{x \to 1^+}\frac{3x^4-8x^3+5}{x^3-x^2-x+1}={-\infty}$$ 
$$\lim_{x \to 1^-}\frac{3x^4-8x^3+5}{x^3-x^2-x+1}={+\infty}$$
The only way I can solve this is using the graph of the function as the limit becomes very apparent. What is the best way to answer this algebraically?

We have
$$
\frac{3x^4-8x^3+5}{x^3-x^2-x+1} = 3x - 5 + \frac{4}{x+1} - \frac{6}{x-1},
$$
which can be found by using polynomial long division and partial fraction decomposition. From this form the limits $x\to 1^+$ and $x\to 1^-$ are easy to compute without looking at any plots.
