Asymptote Calculator

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

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Solution

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: $y=2 x - 1$