Problem: Evaluate the following limit (if it exists)

Solution: We try to substitute infinity. We know (see intuitive evaluation) that when n is large, we can ignore the "−1" and "+1" parts, so we get ln(∞)/∞² = ∞/∞. This is an indeterminate ratio, in fact this is a standard textbook problem that can be solved exactly in the way described in the appropriate box: We pass to functions and apply the l'Hospital rule.

We had to apply the l'Hospital rule twice, which should be no surprise - each application removes one power and we needed to get rid of the square of a logarithm.

It is possible to save some work, since we can pull the square out of the limit (it is a "nice" function):

Then we figure out the limit inside exactly as before, but now one l'Hospital is enough:

Then we have to put the square back: The answer to the given limit is 0² = 0.

In fact, we could have guessed this answer right in the beginning, since we know that "powers beat logarithms", in other words, "the infinity in the denominator is bigger and wins".

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