Tuesday, 5 December 2017

Limit exists or not? limlimitsntoinftyleft[nfracneleft(1+frac1nright)nright]



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}



No comments:

Post a Comment

real analysis - How to find lim_{hrightarrow 0}frac{sin(ha)}{h}

How to find \lim_{h\rightarrow 0}\frac{\sin(ha)}{h} without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...