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

Guest Apr 1, 2015

Best Answer 

 #1
avatar+26750 
+5

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

 

Then x1/m = (1 + δ)1/m → 1 + δ/m as δ → 0

Simnilarly x1/n → 1 + δ/n

 

This means (x1/m -1)/(x1/n - 1) → (1 + δ/m - 1)/(1 + δ/n - 1) → (δ/m)/(δ/n) → n/m

 

So the limit of (x1/m -1)/(x1/n - 1) as x → 1 is n/m

.

Alan  Apr 2, 2015
 #1
avatar+26750 
+5
Best Answer

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

 

Then x1/m = (1 + δ)1/m → 1 + δ/m as δ → 0

Simnilarly x1/n → 1 + δ/n

 

This means (x1/m -1)/(x1/n - 1) → (1 + δ/m - 1)/(1 + δ/n - 1) → (δ/m)/(δ/n) → n/m

 

So the limit of (x1/m -1)/(x1/n - 1) as x → 1 is n/m

.

Alan  Apr 2, 2015

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