Determine whether or not the following limit exists, and find its value if it exists: lim
I think the limit of \left(1+\frac{1}{n}\right)^n is e, but I am not sure I can use this or not in the limit calculation. Could you please help me to solve this? Thank you!
Answer
Note that we can write
\begin{align} n-\frac ne\left(1+\frac1n\right)^n&=n-\frac ne e^{n\log\left(1+\frac1n\right)}\\\\ &=n-\frac ne e^{n\left(\frac1n -\frac{1}{2n^2}+O\left(\frac{1}{n^3}\right)\right)}\\\\ &=n-n\left(1-\frac{1}{2n}+O\left(\frac{1}{n^2}\right)\right) \\\\ &=\frac12+O\left(\frac1n\right) \end{align}
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