Kevin Zhang HMU / lim as x approaches 1 (x^(1/m)-1)/(x^(1/n)-1)

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lim as x approaches 1 (x^(1/m)-1)/(x^(1/n)-1)

Guest Apr 1, 2015

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Best Answer

Let x = 1 + δ, where δ is very small and tends to zero as x tends to 1.

Then x^1/m = (1 + δ)^1/m → 1 + δ/m as δ → 0

Simnilarly x^1/n → 1 + δ/n

This means (x^1/m -1)/(x^1/n - 1) → (1 + δ/m - 1)/(1 + δ/n - 1) → (δ/m)/(δ/n) → n/m

So the limit of (x^1/m -1)/(x^1/n - 1) as x → 1 is n/m

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Alan Apr 2, 2015

Alan · Answer 1 · 2015-04-02T08:08:22

Let x = 1 + δ, where δ is very small and tends to zero as x tends to 1.

Then x^1/m = (1 + δ)^1/m → 1 + δ/m as δ → 0

Simnilarly x^1/n → 1 + δ/n

This means (x^1/m -1)/(x^1/n - 1) → (1 + δ/m - 1)/(1 + δ/n - 1) → (δ/m)/(δ/n) → n/m

So the limit of (x^1/m -1)/(x^1/n - 1) as x → 1 is n/m

.