$$\lim_{x\to a}{\ln{f(x)}}=\ln(\lim_{x\to a}f(x))$$
or
$$\lim_{x\to a}{e^{f(x)}}=e^{(\lim_{x\to a}x)}$$
I always found out those properties when proving limits for the base definition of the number $e$. Does this hold true for other types of functions, such as trigonometric functions? Also how does one make a rigourous proof for the above properties?
If function $g$ is continuous at $a$, then $$ \lim_{x\to a}g(x)=g(\lim_{x\to a}x) $$ $\ln$, $\exp$, $\sin$, $\cos$, and so on are continuous on their domains.