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?