What is the lim_(xrarr1^+) x^(1/(1-x)) as x approaches 1 from the right side?
1 Answer
Jun 14, 2017
1/e
graph{x^(1/(1-x)) [-2.064, 4.095, -1.338, 1.74]}
Well, this would be much easier if we simply took the
ln [lim_(x->1^(+)) x^(1/(1-x))]
= lim_(x->1^(+)) ln (x^(1/(1-x)))
= lim_(x->1^(+)) ln x/(1-x)
Since
= lim_(x->1^(+)) (1"/"x)/(-1)
And of course,
=> ln [lim_(x->1^(+)) x^(1/(1-x))] = -1
As a result, the original limit is:
color(blue)(lim_(x->1^(+)) x^(1/(1-x))) = "exp"(ln [lim_(x->1^(+)) x^(1/(1-x))])
= e^(-1)
= color(blue)(1/e)