40: Putting in infinity easily gives infinity over infinity. How to handle it? First, the square root is on the outside and it is a nice function, so it can be pulled out of the fraction. Since there are just powers in the fraction now, one is tempted to use some tricks from the box "polynomials and ratios with powers".

What can be expected? From intuitive calculations and the scale of powers it follows that the "−1" in the denominator can be ignored and the ratio nⁿ/eⁿ converges to infinity, since the first power dominates the exponential in the denominator. How to prove this guess?

The simplest way to get properly rid of the "−1" is to divide the fraction by eⁿ so that it splits into two parts. The second part will be 1/eⁿ, which goes to zero by the limit algebra and won't bother you any more. The first fraction is the aforementioned nⁿ/eⁿ. How to work it out?

One option is to use the l'Hospital rule, after all it is an indeterminate ratio. Then you can just as well apply l'Hospital to the whole given fraction, the "−1" won't make too much difference. Unfortunately, exponentials do not disappear when differentiated and general powers get even worse, so this is not wise - try it.

Is there another way? Try to use algebra to make this fraction simpler, it might help.

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