# Is $\lim_{x \to x_0} \log(f(x)) = \log\lim_{x \to x_0} f(x)$ always true?

This property is always true? If yes I would like a proof, otherwise an counterexample.

$$\lim\limits_{x \to x_0} \log(f(x)) = \log\lim\limits_{x \to x_0} f(x)$$

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This might be helpful: math.lsa.umich.edu/courses/185F08/composition.pdf – Git Gud Jan 20 '13 at 19:18
This is true since $\log(\cdot)$ is continuous and provided $f(x) > 0$ in a neighborhood of $x_0$ and $\lim_{x \to x_0} f(x) > 0$. – user17762 Jan 20 '13 at 19:22
On the other hand, in the complex numbers no branch of log can be continuous on ${\mathbb C} \backslash \{0\}$, and then it is possible to have $\lim_{z \to z_0} \log f(z) \ne \log \lim_{z \to z_0} f(z)$. – Robert Israel Jan 20 '13 at 19:29

This property is always true because $\log$ is a continuous function.