Asymptoot Rekenmachine
Vind asymptoten stap voor stap
De rekenmachine probeert de verticale, horizontale en scheve asymptoten van de functie te vinden, met uitgewerkte stappen.
Solution
Your input: find the vertical, horizontal and slant asymptotes of the function $$$f(x)=x + \sin{\left(x \right)}$$$
Vertical Asymptotes
The line $$$x=L$$$ is a vertical asymptote of the function $$$y=x + \sin{\left(x \right)}$$$, 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:
$$$\lim_{x \to \infty}\left(x + \sin{\left(x \right)}\right)=\infty$$$ (for steps, see limit calculator).
$$$\lim_{x \to -\infty}\left(x + \sin{\left(x \right)}\right)=-\infty$$$ (for steps, see limit calculator).
Thus, there are no horizontal asymptotes.
Slant Asymptotes
Line $$$y=mx+b$$$ is a slant asymptote of the function $$$y=f{\left(x \right)}$$$, if either $$$m=\lim_{x \to \infty}\left(\frac{f{\left(x \right)}}{x}\right)=L$$$ or $$$m=\lim_{x \to -\infty}\left(\frac{f{\left(x \right)}}{x}\right)=L$$$, and $$$L$$$ is finite and nonzero.
Calculate the first limit:
$$$\lim_{x \to \infty}\left(\frac{x + \sin{\left(x \right)}}{x}\right)=1$$$ (for steps, see limit calculator).
Since the value of the limit is finite and nonzero, then there is a slant asymptote with $$$m=1$$$.
To calculate $$$b$$$, find $$$b=\lim_{x \to \infty}\left(- m x + f{\left(x \right)}\right)=\lim_{x \to \infty} \sin{\left(x \right)}=\text{NaN}$$$ (for steps, see limit calculator).
Therefore, the slant asymptote is $$$y=\text{NaN}$$$.
Calculate the second limit:
$$$\lim_{x \to -\infty}\left(\frac{x + \sin{\left(x \right)}}{x}\right)=1$$$ (for steps, see limit calculator).
Since the value of the limit is finite and nonzero, then there is a slant asymptote with $$$m=1$$$.
But we've already found an asymptote with such a slope.
Answer
No vertical asymptotes.
No horizontal asymptotes.
Slant asymptote: $$$y=\text{NaN}$$$