Find asymptotes of function

Question

I wonder if function $f(x)= 2x\arctan(e^{3x})$ have any asymptotes. I have got $y= \pi x$ for slant and $y=0$ for horizontal asymptote.

Answer 1

We have$$\lim_{x\to +\infty}2x\arctan e^{3x}=\ldots=+\infty,\quad \lim_{x\to -\infty}2x\arctan e^{3x}=\ldots=0$$ so, $y=0$ is an horizontal asymptote on the left side. In the same way, $$m=\lim_{x\to +\infty}\frac{f(x)}{x}=\lim_{x\to +\infty}2\arctan e^{3x}=\pi,\\m=\lim_{x\to -\infty}\frac{f(x)}{x}=\lim_{x\to -\infty}2\arctan e^{3x}=0,$$ and we get $y=\ldots=\pi x+0$ as oblique asymptote on the right side.