Why is Wolfram missing so many oblique asymptotes? (Not only about Wolfram in thread)

Question

Few days ago I made a post, and to be frank I'm not sure if I'm even having this question in the right forum. But I'm also looking for information on if my thoughts are correct.

Observe this little plotted function. Do you agree with me that there should be two and NOT one oblique asymptote? If not, why?

I want to find the constant for the asymptote Wolfram did NOT catch; how can I do this using Wolfram in perhaps a different way? (I do also have Mathematica if that helps).

Answer 1

It looks like Wolfram Alpha only looks for oblique asymptotes as $x\to +\infty$.

You can find the other asymptote by substituting $x\mapsto -x$ by hand and ask for asymptotes of the mirrored function instead.

Answer 2

The function does indeed have another asymptote. I don't know why W|A doesn't provide it.

To find it, simply apply the mapping $x \mapsto -x$, viz. here.