Limit of logarithm

Question

I came across this relation a lot,when evaluating lim of function like natural logarithm I could find that evaluating limit for logarithm equal to evaluating logarithm for limit. Why this is true? $$\lim \ln(f(x))=\ln \lim f(x)$$

Answer 1

This property is continuity, and it's known that the natural logarithm is continuous on its domain.

Put more generally, if $ \lim_{x \to a} g(x) = b $ and if $ f $ is continuous at $ b $, then $ \lim_{x \to a} f(g(x)) = f(\lim_{x \to a} g(x)) $. The natural logarithm is continuous at every point in its domain, so "commuting" the limit with the log is valid.

Answer 2

It follows directly from continuity, as suggested by all other comments. Indeed, if we denote $f_0(x) := \lim(f(x))$, for any $\varepsilon >0$,

$$ \| \ln(f(x)) - \ln(f_0(x)) \| < \varepsilon $$

holds for some $\delta > 0$ such that $ \| f(x) - f_0(x) \| < \delta $. And the latter is implied when $f$ converges to $f_0$.