Oblique asymptote of a rational function

Text

Determine the equation of the oblique asymptote of the function $y = \frac{x^3+x^2+x+1}{2+x-4x^2}.$

Solution

The degree of the numerator ($3$) exceeds that of the denominator ($2$) by one: an oblique asymptote $y=mx+q$ exists. $m = \lim_{x\to\infty}\frac{y}{x} = \lim_{x\to\infty}\frac{x^3+x^2+x+1}{x\,(-4x^2+x+2)} = \frac{1}{-4} = -\frac14.$ $q = \lim_{x\to\infty}\left(y - mx\right) = \lim_{x\to\infty}\frac{(x^3+x^2+x+1) + \frac{x}{4}(-4x^2+x+2)}{-4x^2+x+2} = \lim_{x\to\infty}\frac{\frac54 x^2 + \frac32 x + 1}{-4x^2+x+2} = \frac{5/4}{-4} = -\frac{5}{16}.$ $\boxed{y = -\frac{x}{4} - \frac{5}{16}}$