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This is probably not too smart, just wondering of the name of this rule: $$ \log \lim_{x \to x_0}f(x) = \lim_{x \to x_0}\log f(x) $$ A reference to a source and/or proof would be good too.

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up vote 2 down vote accepted

As long as the function inside the limit symbol is continuous at $x_0$, the statement follows from the continuity of that function. Search for "interchanging limits," and you're sure to find sources.

Edit: Here's something worthwhile from an introductory real analysis textbook.

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It's called continuity. It's the same as saying that the logarithm function is continuous.

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