For f(x) = (x^2+x+1)/x, what is its oblique (slant) asymptote as x approaches infinity?

For f(x) = (x^2+x+1)/x, what is its oblique (slant) asymptote as x approaches infinity?

• A. y = x
• B. y = x + 1
• C. y = 0
• D. y = x - 1

Answer: (B)

When a rational function has a numerator degree one higher than the denominator, it has a slant (oblique) asymptote given by the quotient of the division. Divide x^2 + x + 1 by x: the quotient is x + 1 and the remainder is 1, so f(x) = (x^2 + x + 1)/x = x + 1 + 1/x. As x → ∞, the term 1/x → 0, so f(x) approaches the line y = x + 1. That line is the oblique asymptote.

When a rational function has a numerator degree one higher than the denominator, it has a slant (oblique) asymptote given by the quotient of the division. Divide x^2 + x + 1 by x: the quotient is x + 1 and the remainder is 1, so

f(x) = (x^2 + x + 1)/x = x + 1 + 1/x.

As x → ∞, the term 1/x → 0, so f(x) approaches the line y = x + 1. That line is the oblique asymptote.