# Asymptote Calculator

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

Enter a function: f(x)=

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Your input: find the vertical, horizontal and slant asymptotes of the function $$f(x)=\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{x^{2} + 8 x + 15}$$$## 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 doesn't exist.

So, find the points where the denominator equals $$0$$$and check them. $$x=-5$$$, check:

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

$$x=-3$$$, check: $$\lim_{x \to -3^+}\left(\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{\left(x + 3\right) \left(x + 5\right)}\right)=\infty$$$ (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 the 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: $$y=2 x - 1$$\$