Using L'Hopital's rule, we know that lim_(x->a)(f(x))/(g(x))=>(f'(a))/(g'(a))
f(x)=sqrt(1+x^2)-sqrt(1+x)
=(1+x^2)^(1/2)-(1+x)^(1/2)
f'(x)=x(1+x^2)^(-1/2)-(1+x)^(-1/2)/2
g(x)=sqrt(1+x^3)-sqrt(1+x)
=(1+x^3)^(1/2)-(1+x)^(1/2)
g'(x)=(3x^2(1+x^3)^(-1/2))/2-(1+x)^(-1/2)/2
lim_(x->0)(sqrt(1+x^2)-sqrt(1+x))/(sqrt(1+x^3)-sqrt(1+x))=>(0(1+0^2)^(-1/2)-(1+0)^(-1/2)/2)/((3(0)^2(1+0^3)^(-1/2))/2-(1+0
)^(-1/2)/2)
=(-(1+0)^(-1/2)/2)/(-(1+0)^(-1/2)/2)
=cancel(-(1+0)^(-1/2)/2)/cancel(-(1+0)^(-1/2)/2)=1