Problem: Evaluate the following limit (if it exists)
Solution: We try to substitute infinity.
We see that the first part is fine, but the second needs some work. We obtained an indeterminate ratio, for which we have a box. We can guess the answer, see intuitive evaluation in Theory - Limits, power beaths logarithms so the result is zero. But such an answer is not always sufficient, we need to show some calculations. Here the way seems clear, We split the limit into two parts to be handled separately, then change the second part into functions and apply the l'Hospital rule.
What happens if we get seduced into putting the terms together? We get
If we had only the popular version of the l'Hospital rule, we would have to prove now that the numerator goes to infininty. This is true, since powers beat logarithms (see the scale of powers and Intuitive evaluation in Theory - Limit), and it can be proved by factoring, see the box polynomials and ratios with powers.
Fortunately, we also have a version of the l'Hospital rule for "something over infinity", so we can apply it right away. Check that it leads to the same answer as above.