\lim_{x \to 0} { x + {\sin x} \over x - {\sin x} }=lim_{x \to 0} {( x + {\sin x})' \over (x - {\sin x})'}= \lim_{x \to 0} { 1 + {\cos x} \over 1 - {\cos x} }=\lim_{x \to 0} {(1 + {\cos x})' \over (1 - {\cos x)'}}=lim_{x \to 0}{{- \sin x} \over \sin x}= lim_{x \to 0}{( - \sin x)' \over (\sin x)'}= lim_{x \to 0}{- \cos x \over \cos x}= -1