Monday, 9 January 2017

calculus - Calculate the limit limxrightarrow0dfracefrac1xxn




Let n>0 be an integer. Calculate the limit lim



The limit is of the form \dfrac{0}{0}. Using L'Hospital, the derivative of the denominator is nx^{n-1}, while the derivative of the numerator is \dfrac{e^{-\frac1x}}{x^2}, so that the new fraction is \dfrac{e^{-\frac{1}{x}}}{nx^{n+1}}, which is \dfrac00 again. It doesn't help much.


Answer



Indeed, L'Hôpital fails here. You want x\to 0^+. What you can argue is that is is equivalent to showing \lim\limits_{t\to +\infty}t^n e^{-t}=0 for any n. And L'Hôpital works nicely in this case.


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