How do you find the limit of $rootx(x)$ as $x>oo$ ?

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Answer 1 · 2016-12-29T11:42:55

$lim_(x->oo) root(x)x =1$

We can write $f(x) = root(x)x$ as:

$f(x) = root(x)x = (x^(1/2))^(1/x)=x^(1/(2x))=e^(lnx/(2x))$

so, as:

$lim_(x->oo) lnx/(2x)=0$

$lim_(x->oo) root(x)x = lim_(x->oo)e^(lnx/(2x))= e^0=1$

graph{root(x)x [-0.75, 19.25, -4.88, 5.12]}

How do you find the limit of rootx(x)rootx(x) as x>oox>oo?