Question

Evaluate the following limit
lim as x approaches 0 from the right
(1/x) integral [1,(1-x)] of t^(ln(1-t)) dt

Answer 1

L'hopitals rule!!

Answer 2

lim = 1

case for x=.0001
http://www.wolframalpha.com/input/?i=integrate+t%5Elog%281-t%29+dt+from+0.9999+to+1

Answer 3

\[\lim_{x\to0^+}\frac{\int_1^{1-x}t^{\ln(1-t)}~dt}{x}\]
Upon substituting \(x=0\), you have
\[\lim_{x\to0^+}\frac{\int_1^{1}t^{\ln(1-t)}~dt}{0}=\frac{0}{0}\]
Apply L'Hopital's rule.
\[\lim_{x\to0^+}\frac{\frac{d}{dx}\int_1^{1-x}t^{\ln(1-t)}~dt}{\frac{d}{dx}x}\]
By the FTC, you have
\[\lim_{x\to0^+}\frac{-(1-x)^{\ln(1-(1-x))}}{1}\]