What is the vertical asymptote of $y=2x-\arccos(\frac{1}{x})$?

Question

I have to find the vertical asymptote of $y=2x-\arccos(\frac{1}{x})$. So I have to find the limit of the function when $x$ approaches zero. In my textbook it says that the vertical asymptote does not exist for this function ..why? For $x\to0$ the function goes to infinity so $x=0$ should be a vertical asymptote?

Answer 1

The cosine of any angle is between $-1$ and $1$, so if $|x|<1$, $\arccos\frac1x$ is undefined. Thus, the function $$f(x)=2x-\arccos\frac1x$$ is undefined in the neighborhood $(-1,1)$ of $0$ and cannot have a limit as $x\to 0$.