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?

1 Answer
Jan 22, 2018

-1/32

Explanation:

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