$\displaystyle \dfrac{\lim}{x \to \infty}\dfrac{x^{2} - 4}{2 + x - 4x^{2}}$

Now divide everything by the highest index in the denominator

$\displaystyle \dfrac{\lim}{x \to \infty}\dfrac{x^{2}/ x^{2} - 4/x^{2}}{2/ x^{2}+ x/x^{2} - 4x^{2}/x^{2}}$

$\displaystyle \dfrac{\lim}{x \to \infty}\dfrac{1 -0}{0 + x^{-1} - 4}$ -

See the $\displaystyle x^{-1}$ in the bottom?

$\displaystyle \dfrac{\lim}{x \to \infty}\dfrac{1 - 0 + x^{1}}{0 + - 4}$ -

This becomes $\displaystyle x^{1}$ in the top.

But how to evaluate to get $\displaystyle -\dfrac{1}{4}$