Question Number 85129 by Rio Michael last updated on 19/Mar/20
![lim_(x→e) [∫_0 ^e ((1/x))dx] =?](https://www.tinkutara.com/question/Q85129.png)
$$\underset{{x}\rightarrow{e}} {\mathrm{lim}}\:\left[\underset{\mathrm{0}} {\overset{{e}} {\int}}\left(\frac{\mathrm{1}}{{x}}\right){dx}\right]\:=? \\ $$
Commented by john santu last updated on 19/Mar/20
![∫ _0^e ((1/x)) dx = lim_(x→0) [ ln(x)]_x ^e = lim_(x→0) (1−ln(x)) lim_(x→e) [ lim_(x→0) (1−ln(x)) ] = lim_(x→0) [ lim_(x→e) (1−ln(x) ] = lim_(x→0) (0) = 0](https://www.tinkutara.com/question/Q85133.png)
$$\int\:_{\mathrm{0}} ^{\mathrm{e}} \:\left(\frac{\mathrm{1}}{\mathrm{x}}\right)\:\mathrm{dx}\:=\:\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left[\:\mathrm{ln}\left(\mathrm{x}\right)\right]_{\mathrm{x}} ^{\mathrm{e}} \:=\: \\ $$$$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{1}−\mathrm{ln}\left(\mathrm{x}\right)\right) \\ $$$$\underset{{x}\rightarrow\mathrm{e}} {\mathrm{lim}}\:\left[\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{1}−\mathrm{ln}\left(\mathrm{x}\right)\right)\:\right]\:=\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\:\underset{{x}\rightarrow\mathrm{e}} {\mathrm{lim}}\:\left(\mathrm{1}−\mathrm{ln}\left(\mathrm{x}\right)\:\right]\:=\:\right. \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{0}\right)\:=\:\mathrm{0} \\ $$
Commented by Rio Michael last updated on 19/Mar/20
$$\mathrm{thanks}\:\mathrm{sir} \\ $$