Math · College algebra · Worked example
Find a slant asymptote by division
Find the asymptotes of f(x) = (x² + 1)/(x − 1).
Compare the degrees
The numerator has degree 2 and the denominator degree 1, one less, so there is a slant asymptote instead of a horizontal one.
Divide
x² + 1 = (x − 1)(x + 1) + 2, so f(x) is x + 1 plus a remainder term 2/(x − 1).
Read the slant asymptote
As x grows large in either direction, 2/(x − 1) shrinks toward 0, so the graph approaches the line y = x + 1.
Read the vertical asymptote and intercepts
The denominator is zero at x = 1 and the numerator is 2 there, so x = 1 is a vertical asymptote. x² + 1 is never zero, so there is no x-intercept; the y-intercept is f(0) = −1.
Result
Slant asymptote y = x + 1 and vertical asymptote x = 1.
Your turn
Find the slant asymptote of (2x² − 3x + 1)/(x + 1).
Show the answer and explanation
y = 2x − 5.
Divide: 2x² − 3x + 1 = (x + 1)(2x − 5) + 6, so the function is 2x − 5 + 6/(x + 1). The remainder term fades far out, leaving y = 2x − 5.
Keep exploring
In Graph, trace far to the right: at x = 101 the curve is 2/100 = 0.02 above the line y = x + 1.
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