If #lim_(x->0)(1/x^3)(1/sqrt(1+x)-(1+ax)/(1+bx))# exist, find the values of a, b. How can i solve this?

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Answer 1 · 2018-01-22T10:55:45

#-1/32#

Developing in series

#f(x)=1/sqrt[1 + x] - (1 + a x)/(1 + b x)# near #x=0# we obtain

#f(x) = (b-a-1/2) x + (3/8 + a b - b^2) x^2 + (b^3-5/16 - a b^2) x^3 + O(x^4)# then

#lim_(x->0)(1/x^3)(1/sqrt(1+x)-(1+ax)/(1+bx)) = lim_(x->0) 1/x^3 f(x)#

now solving for #a,b#

#{ (b-a-1/2=0),(3/8 + a b - b^2=0):}#

we obtain

#a= 1/4, b = 3/4# and then

#lim_(x->0)(1/x^3)(1/sqrt(1+x)-(1+ax)/(1+bx))= b^3-5/16 - a b^2 = -1/32#