Given the function, $f:\mathbb{R} \setminus {\{0}\}\to\mathbb{R}$, $f(x)=x+\frac{1}{x}$.
Find the limits of $f$ at infinity(and negative infinity) and all asymptotes of $f$.
I don't know where I have to start. Really, I have no idea.
Given the function, $f:\mathbb{R} \setminus {\{0}\}\to\mathbb{R}$, $f(x)=x+\frac{1}{x}$.
Find the limits of $f$ at infinity(and negative infinity) and all asymptotes of $f$.
I don't know where I have to start. Really, I have no idea.
We can see where is the horizontal asymptote by doing $\lim\limits_{x\to \infty}x+\frac{1}{x}$ and $\lim\limits_{x\to -\infty}x+\frac{1}{x}$
The vertical asymptote is where $\frac{1}{x}=0$ and to the limits with that number. In this case, $\infty \text{ and }-\infty$.
And the oblique asymptotes are given by $f(x)=mx+n+\frac{1}{x}$. So the oblique asymptote in this case is $\\$ $y=x$.
So, the graph is
Try seeing the limit of the difference of this function and $f(x)=x$. Since $\dfrac{1}{x}\rightarrow 0$ when $x\rightarrow\pm\infty$, what can you tell since the limit of the difference is zero?