How do you find $lim sqrt(x+2)-sqrtx$ as $x->oo$ ?

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Answer 1 · 2017-01-22T22:53:43

$lim_(xrarroo)(sqrt(x+2)-sqrtx)$ has indeterminate initial form $oo-oo$ . See below.

$(sqrt(x+2)-sqrtx) = ((sqrt(x+2)-sqrtx))/1 ((sqrt(x+2)+sqrtx))/((sqrt(x+2)+sqrtx))$

$= (x+2-x)/(sqrt(x+2)+sqrtx)$

$= 2/(sqrt(x+2)+sqrtx)$

As $xrarroo$ , the denominator $sqrt(x+2)+sqrtx rarr oo$

Therefore,

$lim_(xrarroo) (sqrt(x+2)-sqrtx) =lim_(xrarroo) 2/(sqrt(x+2)+sqrtx)$

$= 0$

How do you find lim sqrt(x+2)-sqrtxlim sqrt(x+2)-sqrtx as x->oox->oo?