Solving a limit without using L'hospital's rule and series: lim_{x->0} (ln(1+x) - x)/(x sin(x))

[Rene]

Hello everyone, I've been struggling in this limit for a day. Is there any interesting way to solve it without using L'hospital's rule and series?

[math]\lim _ { x \rightarrow 0 } \frac { \ln ( 1 + x ) - x } { x \sin x }[/math]
the attachment is my steps I've been doing so far, thanks!!