Tuesday, 27 March 2018

calculus - Limit Proof of ex/xn




I am wondering how to prove lim



I was thinking of using L'Hospital's rule? But then not sure how to do the summation for doing L'Hospital's rule n times on the denominator? Or whether it would be easier using longs like \lim_{x\to \infty} \ln(e^x)-\ln(x^n)?




Thank you!


Answer



You can certainly use L'Hopital's n times. That is, for each n\geq 0 we have \lim_{x\to\infty}\frac{e^x}{x^n}=\lim_{x\to\infty}\frac{e^x}{nx^{n-1}}=\cdots=\lim_{x\to\infty}\frac{e^x}{n!}=\infty since at each stage we are in \frac{\infty}{\infty} indeterminate form.


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