Indeterminate Forms and L'Hospital's Rule

Question

Find the limit as x approaches 1 from the right 

 

x^(1/(x-1))

Answer 1

I suggest this approach to get an indeterminate form so we can apply L'Hopital's rule:

 

y = x^1/(x-1)

 

Take ln on both sides.

 

ln(y) = [1 / (x - 1)] ln(x) = ln(x) / (x - 1)

 

The right hand side is indeterminate as x→1⁺.  Apply L'Hopital's rule.

 

lim_x→1+ ln(y) = lim_x→1+ [ln(x) / (x - 1)] = lim_x→1+ (1/x) = 1

 

Then ln(y) → 1 as x → 1⁺.

 

Take e⁽⁾ on both sides.

 

y → e¹ as x → 1⁺

 

y = x^1/(x-1)

 

Hence, 

 

x^1/(x-1) → e as x → 1⁺.

 

Test this at 2 numbers very close to 1 but > 1, as in Michael's solution.

 

1.001^1/(1.001-1) ≅ 2.716924

 

1.000001^{1/(1.000001-1)} ≅ 2.718280

 

e ≅ 2.718282

 

They are very close.

 

 

Answer 2

Evaluate the expression at x= 1.001 and at x=1.000001.  If the result reaches a certain constant, then there is a limit.  If not, then limit is infinity.