Is \(\displaystyle\lim_{{{x}\to\infty}}{\frac{{{\ln{{\left\lbrace{x}\right\rbrace}}}}}{{{x}}}}=\lim_{{{x}\to\infty}}{\frac{{{\frac{{{1}}}{{{x}}}}}}{{{1}}}}\) In my notes its given

Breanna Fisher

Breanna Fisher

Answered question

2022-04-15

Is limxln{x}x=limx1x1
In my notes its given limxln{x}x=limx1x1
Is that correct? How do I get that?
I think another example is also related
x1x=eln{x}x

Answer & Explanation

Gonarsu2dw8

Gonarsu2dw8

Beginner2022-04-16Added 19 answers

Both of your example are correct.
Please note that lnx as x. Therefore,
limx=lnxx,
an indeterminate form. Hence by L' Hospital rule,
limx=lnxx=ddx(lnx)ddx(x)=1x1
Your second example: we have x1x=elnx1x=elnxx

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