Lim of (1-b+b(a)^(1/x))^(x) wen x tends 2wrds infinity wid a>0...give method too and hurry wid d answer plz?

Question

Lim of (1-b+b(a)^(1/x))^(x) wen x tends 2wrds infinity wid a>0...give method too and hurry wid d answer plz?

Answer 1

a^b.  

lim x * [a^(1/x) -1]  = log a.

       So, the monstrosity within the bracket is  

1+ b*(a^x-1) =  1+ b{ (log a)/x  + o(x) } as x->inf.

that is,    1+ b*log a /x  + o(x) = 1+ [log(a^b)]/x +o(x).

          Now, if you remember that  

           lim ( 1 + k/x) ^x    = e^k ,

the limit is e^(log(a^b)) = a^b.  

( o(x) doesn't affect the limit - it's order is  1/x^2 . ).

                    Sorry, I'm also in a hurry. A rigorous proof needs taking logarithms & L' Hopital's rule of differentiation.