$\displaystyle \lim_{x\to 0} \frac{2x^2-3x-4}{x^2-1}$

I had this problem on a quiz today and I don't feel comfortable with how I approached it. Since the highest degree of both the variables in the numerator and denominator are the same, I assumed that the limit as x approaches zero would be the quotient of their coefficients, 2. My professor teaches this method of limit taking much more abstractly though when direct substitution doesn't apply, and that's where I am confused. He'll use notation like $\displaystyle \frac{1}{\infty}$ or $\displaystyle -\infty-\infty$ to find the limit of a problem where direct substitution doesn't apply.

For the problem in this post I tried taking a difference of squares in the denominator and factoring the top but that didn't get me anywhere. I guess my question is whether or not anyone's familiar with the abstraction process for indeterminate forms; e.g., $\displaystyle \frac{1}{0}$ being equal to infinity, and if the right answer here is 2?