Kalkulator asymptot
Znajdowanie asymptot krok po kroku
Kalkulator spróbuje znaleźć asymptoty pionowe, poziome i ukośne funkcji, z pokazanymi krokami.
Solution
Your input: find the vertical, horizontal and slant asymptotes of the function f(x)=2x3+15x2+22x−11x2+8x+15
Vertical Asymptotes
The line x=L is a vertical asymptote of the function y=2x3+15x2+22x−11x2+8x+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 (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.
Answer
Vertical asymptotes: x=-5; x=-3
No horizontal asymptotes.
Slant asymptote: y=2 x - 1