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Asymptotenrechner
Asymptoten Schritt für Schritt finden
Der Rechner versucht, die vertikalen, horizontalen und schiefen Asymptoten der Funktion zu bestimmen; die Schritte werden angezeigt.
Solution
Your input: find the vertical, horizontal and slant asymptotes of the function $$$f(x)=\frac{1}{x}$$$
Vertical Asymptotes
The line $$$x=L$$$ is a vertical asymptote of the function $$$y=\frac{1}{x}$$$, 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.
$$$x=0$$$, check:
$$$\lim_{x \to 0^+} \frac{1}{x}=\infty$$$ (for steps, see limit calculator).
Since the limit is infinite, then $$$x=0$$$ 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:
$$$\lim_{x \to \infty} \frac{1}{x}=0$$$ (for steps, see limit calculator).
$$$\lim_{x \to -\infty} \frac{1}{x}=0$$$ (for steps, see limit calculator).
Thus, the horizontal asymptote is $$$y=0$$$.
Slant Asymptotes
Since the degree of the numerator is not one degree greater than the denominator, then there are no slant asymptotes.
Answer
Vertical asymptote: $$$x=0$$$
Horizontal asymptote: $$$y=0$$$
No slant asymptotes.