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Answer 1 · 2017-09-30T18:03:10

$lim_(xrarroo)sqrt(x+sqrt(x+sqrtx))/sqrt(x+1)=1$

Let's try to factor within the square roots:

$lim_(xrarroo)sqrt(x+sqrt(x+sqrtx))/sqrt(x+1)$

$=lim_(xrarroo)sqrt(x+sqrt(x(1+1/sqrtx)))/sqrt(x(1+1/x))$

Pull out what we've just factored from their respective square roots:

$=lim_(xrarroo)sqrt(x+sqrtxsqrt(1+1/sqrtx))/(sqrtxsqrt(1+1/x))$

Factor from the numerator again:

$=lim_(xrarroo)sqrt(x(1+1/sqrtxsqrt(1+1/sqrtx)))/(sqrtxsqrt(1+1/x))$

And pull this from the numerator:

$=lim_(xrarroo)(sqrtxsqrt(1+1/sqrtxsqrt(1+1/sqrtx)))/(sqrtxsqrt(1+1/x))$

Which cancels:

$=lim_(xrarroo)sqrt(1+1/sqrtxsqrt(1+1/sqrtx))/sqrt(1+1/x)$

Note that both $1/x$ and $1/sqrtx$ approach $0$ as $x$ approaches infinity.

$=sqrt(1+0sqrt(1+0))/sqrt(1+0)$

$=sqrt1/sqrt1$

$=1$