Wrong application of L'Hôpital's rule?
You seem to be assuming that L'Hôpital preserves asymptotes, when it's not the case.
Take for instance $$ \frac{x^2(x+2)}{x^2}, $$ with the obvious asymptote $x+2$. If you take derivatives to use L'Hôpital, you get $$ \frac{3x^2+4x}{2x}=\frac{3x+4}{2}, $$ and the asymptote is not the same.
To find the asymptote:
$$y=mx+n$$
you should calculate separately the following limits:
$$m=\frac{g(x)}{x}$$
for the slope, and
$$n=g(x)-mx$$
for the intercept.
Take also a look here:
How to find the oblique asymptote of root of a function?