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Your input: find the vertical, horizontal and slant asymptotes of the function $$$f(x)=\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{x^{2} + 8 x + 15}$$$

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.

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.

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 $$$y=2 x - 1$$$.

Answer

Vertical asymptotes: $$$x=-5$$$; $$$x=-3$$$

No horizontal asymptotes.

Slant asymptote: $$$y=2 x - 1$$$