Limit of y=((x-1)/x)^x as x approaches inf

[Chris101]

I reasoned that (x-1)/x is always less than 1 for positive x. Therefore it will tend to zero as the exponent tends to infinity. But what is confusing is that when working it out for 1 to 100, the value increases.

Is the the limit 0?, and when does the limit "turn".

 

[Cyosis]

The limit is not 0. Use $\lim_{x \to 0} (1+x)^{\frac{1}{x}}=e$.

 

[n1person]

alternatively you can take the natural log of both sides,

ln y = x ln ((x-1)/x) = ln((x-1)/x)/(1/x)

then apply l'hopital's rule to get a nice simple solution, don't forget to then exponent it to solve for y! :)