title says everything. How do I evaluate the limit given ?
How to evaluate the following limit $\lim_{x\rightarrow \infty} \frac{x^x - (x-1)^x}{x^x}$?
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0@DJC : plotting it in gnuplot :) – 2011-02-10
1 Answers
Perhaps try dividing? Then $ \lim_{x\to\infty}\frac{x^x-(x-1)^x}{x^x}=\lim_{x\to\infty}\left[1-\left(\frac{x-1}{x}\right)^x\right]=\lim_{x\to\infty}\left[1-\left(1-\frac{1}{x}\right)^x\right]. $
Notice $ \lim_{x\to\infty}\left(1-\frac{1}{x}\right)^x=e^{\lim_{x\to\infty}x\ln(1-\frac{1}{x})}.(*) $
Try making a substitution like $u=1/x$ to get a situation in which l'Hôpital's rule applies to find this limit. Remember this will also change the value the limit approaches. It should look something like this: $ e^{\lim_{u\to 0}\frac{\ln(1-u)}{u}}=e^{\lim_{u\to 0}\frac{1}{u-1}}. $ Apologies for the poor legibility, I hope it at least gets you started.
*Edit: As Sivaram kindly pointed out, you could use the fact that $\lim_{x\to\infty}(1-\frac{1}{x})^x=e^{-1}$ to get the result right off the bat.
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0You k$n$ow... the most i$n$teresting fact is that now I have no idea why I asked it... :) – 2011-11-22