Question

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

Answer 1

`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`