One of the conditions of applying L'Hospital's Rule is that $f'(x)/g'(x)$ must exist.
$$\lim\limits_{x \to \infty} \frac{f'(x)}{g'(x)}$$

After one application of L'Hospitals, you arrived at a finite numerator over a finite denominator. But while the numerator was finite, it was non-convergent, and so that limit did not exist. Just like the much simpler $\sin(x)$ does not converge to a single value when x approaches infinity -- it oscillates between +/-1. $$\lim\limits_{x \to \infty} {\sin(x)}$$

So all of L'Hospital's pre-conditions must exist for you to use it. As others have mentioned, this limit could more easily be solved by using the squeezing theorem. The numerator's value gets squeezed between $x+1$ and $x-1$. Both of those limits go to $1/5$.