Saturday, 26 August 2017

calculus - Find the limit : limlimitsnrightarrowinftyintn+7nfracsinxx,mathrmdx



I have this exercise I don't know how to approach :





Find the limit : lim




I can see that with n\rightarrow\infty the area under the graph of this function becomes really small as \sin{x} \leq 1 so \dfrac{\sin{n}}{n}\rightarrow_{\infty}0 but can I get something from it?


Answer



Hint: \def\abs#1{\left|#1\right|}
\abs{\int_n^{n+7}\frac{\sin x} x \, dx}\le \int_n^{n+7}\frac{\abs{\sin x}}{x}\, dx\le \frac 1n \int_n^{n+7}\abs{\sin x}\, dx


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