# Asymptote Calculator

The calculator will find the vertical, horizontal and slant asymptotes of the function, with steps shown.

• In general, you can skip the multiplication sign, so 5x is equivalent to 5*x.
• In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x).
• Also, be careful when you write fractions: 1/x^2 ln(x) is 1/x^2 ln(x), and 1/(x^2 ln(x)) is 1/(x^2 ln(x)).
• If you skip parentheses or a multiplication sign, type at least a whitespace, i.e. write sin x (or even better sin(x)) instead of sinx.
• Sometimes I see expressions like tan^2xsec^3x: this will be parsed as tan^(2*3)(x sec(x)). To get tan^2(x)sec^3(x), use parentheses: tan^2(x)sec^3(x).
• Similarly, tanxsec^3x will be parsed as tan(xsec^3(x)). To get tan(x)sec^3(x), use parentheses: tan(x)sec^3(x).
• From the table below, you can notice that sech is not supported, but you can still enter it using the identity sech(x)=1/cosh(x).
• If you get an error, double check your expression, add parentheses and multiplication signs where needed, and consult the table below.
• All suggestions and improvements are welcome. Please leave them in comments.
The following table contains the supported operations and functions:
 Type Get Constants e e pi pi i i (imaginary unit) Operations a+b a+b a-b a-b a*b a*b a^b, a**b a^b sqrt(x), x^(1/2) sqrt(x) cbrt(x), x^(1/3) root(3)(x) root(x,n), x^(1/n) root(n)(x) x^(a/b) x^(a/b) abs(x) |x| Functions e^x e^x ln(x), log(x) ln(x) ln(x)/ln(a) log_a(x) Trigonometric Functions sin(x) sin(x) cos(x) cos(x) tan(x) tan(x), tg(x) cot(x) cot(x), ctg(x) sec(x) sec(x) csc(x) csc(x), cosec(x) Inverse Trigonometric Functions asin(x), arcsin(x), sin^-1(x) asin(x) acos(x), arccos(x), cos^-1(x) acos(x) atan(x), arctan(x), tan^-1(x) atan(x) acot(x), arccot(x), cot^-1(x) acot(x) asec(x), arcsec(x), sec^-1(x) asec(x) acsc(x), arccsc(x), csc^-1(x) acsc(x) Hyperbolic Functions sinh(x) sinh(x) cosh(x) cosh(x) tanh(x) tanh(x) coth(x) coth(x) 1/cosh(x) sech(x) 1/sinh(x) csch(x) Inverse Hyperbolic Functions asinh(x), arcsinh(x), sinh^-1(x) asinh(x) acosh(x), arccosh(x), cosh^-1(x) acosh(x) atanh(x), arctanh(x), tanh^-1(x) atanh(x) acoth(x), arccoth(x), cot^-1(x) acoth(x) acosh(1/x) asech(x) asinh(1/x) acsch(x)

Enter a function: f(x)=

Write all suggestions in comments below.

Your input: find the vertical, horizontal and slant asymptotes of the function $$\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{x^{2} + 8 x + 15}$$$. ## Vertical Asymptotes The line $$x=L$$$ is a vertical asymptote of the function $$y=\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{x^{2} + 8 x + 15}$$$, if the limit of the function (one-sided) at this point is infinite. In other words, it means that possible points are points where the denominator equals 0 or does not exist. So, find the points where the denominator equals 0 and check them. $$x=-5$$$, check:

$$\lim_{x \to -5^+}\left(\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{x^{2} + 8 x + 15}\right)=-\infty$$$(for steps see limit calculator). Since the limit is infinite, then $$x=-5$$$ is a vertical asymptote.

$$x=-3$$$, check: $$\lim_{x \to -3^+}\left(\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{x^{2} + 8 x + 15}\right)=\infty$$$ (for steps see limit calculator).

Since the limit is infinite, then $$x=-3$$$is a vertical asymptote. ## Horizontal Asymptotes Line $$y=L$$$ is a horizontal asymptote of the function $$y=f{\left (x \right )}$$$, if either $$\lim_{x \to \infty} f{\left (x \right )}=L$$$ or $$\lim_{x \to -\infty} f{\left (x \right )}=L$$$, and $$L$$$ is finite.

Calculate limits:

$$\lim_{x \to \infty}\left(\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{x^{2} + 8 x + 15}\right)=\infty$$$(for steps, see limit calculator). $$\lim_{x \to -\infty}\left(\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{x^{2} + 8 x + 15}\right)=-\infty$$$ (for steps, see limit calculator).

Thus, there are no horizontal asymptotes.

## Slant Asymptotes

Do polynomial long division $$\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{x^{2} + 8 x + 15}=2 x - 1 + \frac{4}{x^{2} + 8 x + 15}$$$(for steps, see polynomial long division calculator). The rational term approaches 0 as the variable approaches infinity. Thus, the slant asymptote is $$y=2 x - 1$$$.

Vertical asymptotes: $$x=-5$$$; $$x=-3$$$.
Slant asymptote: $$y=2 x - 1=2.0 x - 1.0$$\$.