Question

How do you find the slant asymptote of `(x^2+3x-4)/x`?

Answer 1

Divide the expressions and discard the remainder.

Explanation:

Ok, so the slant asymptote means that as x gets infinitely big, the graph gets closer to a particular line, which is the slant asymptote. So you divide the numerator by the denominator, and, assuming it's ( `x^2`+... ) / ( `x` +....) and ignore any remainder , then you should get a line equation y=mx+b. That line is your slant asymptote.

In this case you can divide all numerator terms by x, so `x+3+4/x`. The x+3 part is your slant asymptote.

Please note that slant asymptotes only occur when the degree (highest exponent) in the numerator is exactly one more than the denominator, in this case 2 & 1. Also note that in a more complicated case, as if you had say (x+1) in the denominator, then you'd have to use long or synthetic division to get the asymptote.