#lim_(x->0)(sqrt(1+x^2)-sqrt(1+x))/(sqrt(1+x^3)-sqrt(1+x))=#?

1 Answer
1s2s2p
Dec 21, 2017

#lim_(x->0)(sqrt(1+x^2)-sqrt(1+x))/(sqrt(1+x^3)-sqrt(1+x))=1#

Explanation:

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#

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