What is the limit of ((1-x)/(1+x))^(1/x) as x approaches 0?

1 Answer
Mar 1, 2018

e^(-2)=1/(e^2)

Explanation:

We know that,
lim_(color(red)(X)->0)(1+color(red)(X))^(1/color(red)(X))=e
And, color(red)(lim_(x->a)[(f(x))/(g(x))]=(lim_(x->a)f(x))/(lim_(x->a)g(x)))
So,
lim_(x->0)((1-x)/(1+x))^(1/x)=(lim_(x->0)(1-x)^(1/x))/(lim_(x->0)(1+x)^(1/x))=((lim_(x->0)(1+(-x))^(-1/x))^-1)/(lim_(x->0)(1+x)^(1/x))=(e^-1)/e=e^-1*e^-1=e^-2