Maybe just confusion on my part over trying to evaluate a limit. If after taking the derivative then you still end up with infinity over infinity I assume you can apply the rule recursively? So does this mean you are eventually guaranteed that you will have a meaningful fraction when you finish
The limit I was doing when trying to figure this out was $n^2 +1 / 2^n$. But the question is generally speaking
On
You may repeat the L'Hospital rule as many times as it applies. There is no guarantee that it eventually works.
Sometimes it is easier to simplify the fraction before applying L'Hospital rule.
For example in case of $$\frac {x^3+\sqrt x}{x^2+1}$$ if you divide top and bottom by $x^2$ you avoid L'Hospital rule.
You may apply L'Hospital's rule again, but there is no guarantee that it will terminate. Just consider $$ \lim_{x\to\infty}\frac{x}{(1+x^2)^{1/2}}. $$ Applying L'Hospital's rule again and again gives $$ \lim_{x\to\infty}\frac{x}{(1+x^2)^{1/2}} = \lim_{x\to\infty}\frac{(1+x^2)^{1/2}}{x} = \lim_{x\to\infty}\frac{x}{(1+x^2)^{1/2}} = \dots, $$ and this procedure never ends.