Slant Asymptote Calculator

A slant (or oblique) asymptote is a non-horizontal line $y = mx + b$ that the graph approaches as $x \to \pm\infty$.

For a rational function $f(x) = \dfrac{P(x)}{Q(x)}$ the condition is exact:

$\deg P = \deg Q + 1$

One degree more — no more, no less. If the degrees are equal you get a horizontal asymptote; if the numerator is smaller you get $y = 0$; if it exceeds the denominator by two or more, the end behaviour follows a curve (parabolic or higher), not a line.

A function never has a slant asymptote and a horizontal one, since the two describe the same limit. It can, however, have a slant asymptote together with several vertical asymptotes. Reduce the fraction first: cancelling a common factor can change the degrees and therefore the answer.

The method

1. Check the degree condition $\deg P = \deg Q + 1$. If it fails, stop — there is no slant asymptote.
2. Divide $P(x)$ by $Q(x)$ with polynomial long division (synthetic division works when the divisor is linear).
3. Write the result as

$f(x) = \underbrace{mx + b}_{\text{quotient}} + \frac{R(x)}{Q(x)}$

4. Discard the remainder term. Because $\deg R < \deg Q$, the fraction $R(x)/Q(x) \to 0$ as $x \to \pm\infty$.
5. The slant asymptote is the quotient: $y = mx + b$.

Why the remainder can be dropped

That vanishing remainder is the whole justification, and it also tells you which side the curve approaches from. If $R(x)/Q(x) > 0$ for large positive $x$, the graph sits above the line there; if negative, below. Evaluating the remainder term at one large value is enough to sketch it.

Limit version

Equivalently $m = \lim\limits_{x \to \infty} \dfrac{f(x)}{x}$ and $b = \lim\limits_{x \to \infty} \left[f(x) - mx\right]$, which works for non-rational functions too.

• Skipping the degree check: dividing a function whose degrees differ by two produces a parabola, not a line. That is a curvilinear asymptote, and calling it slant is wrong.
• Keeping the remainder: the asymptote is the quotient only. Writing $y = x + 4 + \frac{6}{x-1}$ describes the function, not its asymptote.
• Forgetting missing terms in the division: write $x^3 + 2$ as $x^3 + 0x^2 + 0x + 2$ before dividing, or every column shifts.
• Cancelling too late: simplify the fraction first, since a common factor can drop the numerator's degree and remove the slant asymptote entirely.
• Assuming the graph stays on one side: it can cross the slant asymptote wherever $R(x) = 0$.