Use the following identity?

Question

Use the following identity aⁿ-1 = (a^n-1+ a^a-2 +...+ a+1) to determine the limit of:

lim x -> 1 [(x^(1/n)-1)/ (x^(1/m)-1)]

Answer 1

Note that x^mn -> 1 as x -> 1. Therefore

lim(x->1) [x^1/n - 1]/[x^1/m - 1] = lim(x->1) [(x^mn)^1/n - 1]/[(x^mn)^1/m - 1] = lim(x->1) (x^m - 1)/(xⁿ - 1)

............................................= lim(x ->1) [(x - 1)(x^m-1 + x^m-2 + ••• x + 1)]/[(x - 1)(x^n-1 + x^n-2 + ••• + x + 1)]

............................................= lim(x->1) (x^m-1 + x^m-2 + ••• + x + 1)/(x^n-1 + x^n-2 + ••• + x + 1)

............................................= m/n