How can I approximate these two functions?
$\sin{(x+\log(x+5))}= \sin{(x)}\cos{\log{(x+5)}}+\cos{x}\sin{(\log{(x+5)})}$ by the trig identity.
It's limit as $x\to 0$ is $\sin{(\log(x+5))}$
Assuming going forward that $log$ has base $e$...
Taylors' might be applicable if you start with the reciprocal of the log's argument.
$\ln{(x+5)}=-\ln{(\frac{1}{x+5})}\approx-\ln{(1-4/5)}\approx-1\cdot(-4/5)$
since $\ln(1-x)\approx -x$
$\sin{(4/5)} \approx 4/5-\frac{(4/5)^3}{3!}$