Let f(x)=|x|+1 and g(x)=[x]-1 , where [.] is the greatest integer...

and $$\begin{aligned} f(x) & =|x|+1 \\ g(x) & =[x]-1 \\ h(x) & =\frac{f(x)}{g(x)}=\frac{|x|+1}{[x]-1}\end{aligned}$$\lim _{x \rightarrow 0} h(x)+\lim _{x \rightarrow 0^{+}} h(x)=\lim _{x \rightarrow 0^{-}} \frac{|x|+1}{[x]-1}\begin{array}{ll}=\lim _{x \rightarrow 0^{-}} \frac{|x|+1}{[x]-1} & \\ =\frac{0+1}{-1-1}+\frac{0+1}{0-1} & \left\{\begin{array}{l}{[0-h]=-1} \\ {[0+h]=0}\end{array}\right\}$\\ =$\frac{1}{2}-$1=-$\frac{3}{2}$ & \end{array}