21: When the limit algebra is tried, the fraction inside the power is "infinity over infinity". Since the power is n and not a constant, it cannot be pulled out of the limit so that the limit would only apply to the fraction - that way the l'Hospital rule could be used. But the power stays inside the limit and l'Hospital is out. What to do?
Since the fraction is just a simple ratio of polynomials, tricks from the box
"polynomials and ratios with
powers" should help determine its limit, only then it will be time to
worry about the power. Why worry? If the limit of the fraction is not 1, then
this number raised to infinity gives an answer by the limit algebra. But if
the fraction goes to 1, then the power is indefinite
How does it look like intuitively? When n grows large, the
So we got lucky and this limit can be done by mere substitution of infinity. It only remains to work out properly the fraction, cancel the leading power (we already saw that it is n).
If you are not sure whether the power is indeed definite and yields zero, you can check it by using the standard trick for powers (see the box "indeterminate power"), that is, use the "e to ln" trick and pass to the limit of the expression inside the exponential.