Asymptotes Calculator - Step-by-Step Solutions

Enter a function of $x$. The calculator finds its vertical, horizontal and oblique (slant) asymptotes, shows step by step how the limits that give them are found, and draws the graph of the function with its asymptotes.

How to enter a function

• Write powers with ^: x^2 is $x^2$, e^(2x) is $e^{2x}$ and 2^x is $2^x$.
• The multiplication sign can be left out: 2x, 3(x+1), x e^(1/x). You can also write 2*x.
• (x^2+1)/(x-2) is $\frac{x^2+1}{x-2}$: put the numerator and the denominator in brackets.
• sqrt(x) is $\sqrt{x}$, cbrt(x) or x^(1/3) is $\sqrt[3]{x}$, and root(x, 4) is $\sqrt[4]{x}$.
• ln(x) is the natural logarithm, log(x) is the logarithm to base 10 and log_2(x) is $\log_2 x$.
• sin(x), cos(x), tan(x), cot(x), arcsin(x), arccos(x), arctan(x), sinh(x), cosh(x), tanh(x).
• |x| is the absolute value, pi is $\pi$ and e is the number $e$.
• You can also type f(x) = ... or y = .... The function may contain one letter only, its variable.

Examples: $\frac{x^2 + 1}{x - 2}$ $\frac{2x^2 - 3x}{x^2 - 4}$ $\frac{x^3}{x^2 - 1}$ $\sqrt{x^2 + 2x}$ $x e^{\frac{1}{x}}$ $\frac{\ln x}{x}$ $\arctan x$ $\tan x$

How to find the asymptotes of a function

An asymptote of a function is a straight line that its graph comes closer and closer to, as the graph runs off to infinity. There are three kinds of asymptotes, and each is found with limits.

Vertical asymptotes

The line ${x = a}$ is a vertical asymptote of $f$ when at least one of the one-sided limits of $f$ at $a$ is infinite:

Such a point is not in the domain of $f$, so look at its ends: the zeros of a denominator, the points where the argument of a logarithm tends to 0, the points where $\tan x$ is not defined. When the limit at such a point is finite, the line is not an asymptote: ${f(x) = \frac{x^2 - 1}{x - 1}}$ has a hole at ${x = 1}$, because ${\frac{x^2 - 1}{x - 1} = x + 1}$ for ${x \neq 1}$.

Horizontal asymptotes

The line ${y = b}$ is a horizontal asymptote as ${x \to +\infty}$ when ${\lim\limits_{x \to +\infty} f(x) = b}$, and as ${x \to -\infty}$ when ${\lim\limits_{x \to -\infty} f(x) = b}$. The two may differ: $\arctan x$ has the horizontal asymptote ${y = \frac{\pi}{2}}$ on the right and ${y = -\frac{\pi}{2}}$ on the left, and $e^x$ has ${y = 0}$ on the left only.

Oblique (slant) asymptotes

The line ${y = kx + b}$ with ${k \neq 0}$ is an oblique asymptote as ${x \to +\infty}$ when ${f(x) - (kx + b) \to 0}$. Its slope and its intercept are two limits:

Both limits must be finite. The same holds as ${x \to -\infty}$. A horizontal asymptote is the case ${k = 0}$, so in each direction a function has a horizontal asymptote, an oblique one, or neither.

Asymptotes of rational functions

For a rational function ${f(x) = \frac{P(x)}{Q(x)}}$, first cancel the common factors of $P$ and $Q$. Then:

• every zero $a$ of the denominator gives a vertical asymptote ${x = a}$;
• when the degree of $P$ is less than the degree of $Q$, the horizontal asymptote is ${y = 0}$;
• when the degrees are equal, the horizontal asymptote is ${y = \frac{a}{b}}$, where $a$ and $b$ are the leading coefficients of $P$ and $Q$;
• when the degree of $P$ is 1 more than the degree of $Q$, divide $P$ by $Q$: the quotient ${kx + b}$ gives the oblique asymptote ${y = kx + b}$;
• when the degree of $P$ is more than 1 greater, there are no horizontal or oblique asymptotes.

For example, ${\frac{x^2 + 1}{x - 2} = x + 2 + \frac{5}{x - 2}}$, so its asymptotes are ${x = 2}$ and ${y = x + 2}$. A rational function has the same horizontal or oblique asymptote as ${x \to +\infty}$ and as ${x \to -\infty}$.

Examples by type of function

Click an example to see the calculator at work.

1. Rational functions. The zeros of the denominator, the degrees, the division. Examples: $\frac{2x^2 - 3x}{x^2 - 4}$, $\frac{x^2 - 1}{x - 1}$, $\frac{x^3 + 2}{x^2 - 1}$
2. Roots. Since ${\sqrt{x^2} = |x|}$, a root can have one asymptote as ${x \to +\infty}$ and another as ${x \to -\infty}$.
   ${\sqrt{x^2 + 2x} = |x| \sqrt{1 + \frac{2}{x}}}$
   Examples: $\sqrt{x^2 + 2x}$, $\frac{x}{\sqrt{x^2 + 1}}$, $x - \sqrt{x^2 - 4}$
3. Exponential functions. ${e^x \to 0}$ as ${x \to -\infty}$ and grows faster than any power as ${x \to +\infty}$. Examples: $\frac{e^x}{x}$, $\frac{1}{1 + e^x}$, $x e^{\frac{1}{x}}$
4. Logarithms. ${\ln x \to -\infty}$ as ${x \to 0^+}$, and it grows slower than any power as ${x \to +\infty}$. Examples: $\frac{\ln x}{x}$, $\ln\frac{x + 1}{x - 1}$, $\frac{1}{\ln x}$
5. Trigonometric functions. $\tan x$ has the vertical asymptotes ${x = \frac{\pi}{2} + k\pi}$, and $\arctan x$ the horizontal asymptotes ${y = \pm\frac{\pi}{2}}$. A function that oscillates, such as $\sin x$, has no limit at infinity. Examples: $\tan x$, $x \arctan x$, $\frac{\sin x}{x}$

Other resources

Function plotter Coordinate planes and graphs Functions and limits Operations on functions Limits Continuous functions How to graph quadratic functions Polynomial long division calculator