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Indeterminate Forms and L'Hospital's Rule

Find the limit as x approaches 1 from the right 
 
x^(1/(x-1))

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Arturo O. | Experienced College Instructor for Physics TutoringExperienced College Instructor for Physi...
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I suggest this approach to get an indeterminate form so we can apply L'Hopital's rule:
 
y = x1/(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.
 
limx→1+ ln(y) = limx→1+ [ln(x) / (x - 1)] = limx→1+ (1/x) = 1
 
Then ln(y) → 1 as x → 1+.
 
Take e() on both sides.
 
y → e1 as x → 1+
 
y = x1/(x-1)
 
Hence, 
 
x1/(x-1) → e as x → 1+.
 
Test this at 2 numbers very close to 1 but > 1, as in Michael's solution.
 
1.0011/(1.001-1) ≅ 2.716924
 
1.0000011/(1.000001-1) ≅ 2.718280
 
e ≅ 2.718282
 
They are very close.
 
 
Michael J. | Effective High School STEM Tutor & CUNY Math Peer LeaderEffective High School STEM Tutor & CUNY ...
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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.