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

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The limit of the function y = ((x-1)/x)^x as x approaches infinity is not zero, contrary to initial assumptions. The expression (x-1)/x approaches 1, leading to the conclusion that the limit equals e, as derived from the limit property \lim_{x \to 0} (1+x)^{\frac{1}{x}}=e. By applying the natural logarithm and L'Hôpital's rule, one can simplify the expression to find the limit definitively.

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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".
 
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The limit is not 0. Use [itex]\lim_{x \to 0} (1+x)^{\frac{1}{x}}=e[/itex].
 
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! :)
 

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