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For example, let's say we have some expression: $$ \lim_{n \rightarrow \infty} \exp(f(n)) $$

This can be solved as $$ \exp(\lim_{n \rightarrow \infty} f(n)) $$

What governs the ability that allows us to move the limit inside the exponential? Based on When can I move the limit operand into a function?, it seems the function (exponential in this case) has to be continuous.

Is that the only requirement? Are there any requirements for things like convexity or monotonicity of the function?

student010101
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  • You can always prove it using the epsilon delta proof or another definition of the limit. – Тyma Gaidash May 01 '21 at 13:43
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    You need two things: 1. $\lim f(n)$ needs to exist and be finite. 2. $\exp$ must be continuous at $\lim f(n)$. Try to prove that you can interchange the limit given these conditions. – Rushabh Mehta May 01 '21 at 13:46

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