What is the limit of ${(1 + 2 x)}^{\frac{1}{x}}$ $(1+2x)^(1/x)$ as x approaches infinity?

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Answer 1 · 2016-08-08T13:24:31

1

$lim_(x to oo) (1+2x)^(1/x)$

$= lim_(x to oo) e^ ( ln (1+2x)^(1/x) )$

$=e^ ( lim_(x to oo) ln (1+2x)^(1/x) )$ as exponential function is continous

$=e^L$

$L = lim_(x to oo) ln (1+2x)^(1/x)$

$= lim_(x to oo) 1/xln (1+2x)$

$= lim_(x to oo) (ln (1+2x))/x$

which is $oo/oo$ indeterminate so we can use L Hopital

$= lim_(x to oo) (1/ (1+2x))/1$

$implies L = lim_(x to oo) 1/ (1+2x) = 0$

and
$e^0 = 1$

What is the limit of (1+2x)^(1/x)(1+2x)1x(1+2x)^(1/x) as x approaches infinity?