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)#