[tex]\displaystyle \mathtt{ \lim_{x \to 0} \frac{x+sin~x}{x^2+sin~x}\overset{\underset{\mathrm{l'H}}{}}{=} \lim_{x \to 0} \frac{(x+sin~x)'}{\left(x^2+sin~x\right)'}=\lim_{x \to 0} \frac{x'+(sin~x)'}{\left(x^2\right)'+(sin~x)'}= }\\ \\ \mathtt{=\lim_{x \to 0} \frac{1+cos~x}{2x+cos~x}= \frac{1+cos~0}{2\cdot0+cos~0}= \frac{1+1}{0+1}= \frac{2}{1}=2 }[/tex]
