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Calcolatore di asintoti
Trova gli asintoti passo dopo passo
Il calcolatore cercherà di determinare gli asintoti verticali, orizzontali e obliqui della funzione, mostrando i passaggi.
Solution
Your input: find the vertical, horizontal and slant asymptotes of the function $$$f(x)=\frac{x^{3} - 2 x^{2}}{x^{2} + 1}$$$
Vertical Asymptotes
The line $$$x=L$$$ is a vertical asymptote of the function $$$y=\frac{x^{3} - 2 x^{2}}{x^{2} + 1}$$$, 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.
As can be seen, there are no such points, so this function doesn't have vertical asymptotes.
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:
$$$\lim_{x \to \infty}\left(\frac{x^{3} - 2 x^{2}}{x^{2} + 1}\right)=\infty$$$ (for steps, see limit calculator).
$$$\lim_{x \to -\infty}\left(\frac{x^{3} - 2 x^{2}}{x^{2} + 1}\right)=-\infty$$$ (for steps, see limit calculator).
Thus, there are no horizontal asymptotes.
Slant Asymptotes
Do polynomial long division $$$\frac{x^{3} - 2 x^{2}}{x^{2} + 1}=x + \frac{2 - x}{x^{2} + 1} - 2$$$ (for steps, see polynomial long division calculator).
The rational term approaches 0 as the variable approaches infinity.
Thus, the slant asymptote is $$$y=x - 2$$$.
Answer
No vertical asymptotes.
No horizontal asymptotes.
Slant asymptote: $$$y=x - 2$$$