4: You should get

Note that once the ln(0) appeared, it was necessary to investigate the zero closer. Going back to where it started, from n positive one gets that 1 + 1/n > 1, so ln(1 + 1/n) > 0, that is, it gives a "positive zero" 0⁺.

The new limit expression is an indeterminate ratio, and since no tricks are apparent, do it the standard way: Use the l'Hospital rule.

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