Prove that if limh0g(h)=0 and g(h)0 for some interval (ˉδ,0)(0,ˉδ) where ˉδ>0 then limh0g(h)=limg(h)0g(h).

Problem: 

Prove that if limh0g(h)=0 and g(h)0 for some interval (ˉδ,0)(0,ˉδ) where ˉδ>0 then
limh0g(h)=limg(h)0g(h).

Answer: 

limh0g(h)=limg(h)0g(h).