How do you find $lim sqrt(2x+3)-sqrtx$ as $x->oo$ ?

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Answer 1 · 2018-01-18T16:17:16

$lim_(xrarr+oo)(sqrt(2x+3)-sqrtx)=+oo$

$lim_(xrarr+oo)(sqrt(2x+3)-sqrtx)=?$

$f(x)=sqrt(2x+3)-sqrtx=$

$((sqrt(2x+3)-sqrtx)(sqrt(2x+3)+sqrtx))/(sqrt(2x+3)+sqrtx)$ $=$

$(sqrt(2x+3)^2-sqrtx^2)/(sqrt(2x+3)+sqrtx)$ $=$

$(2x+3-x)/(sqrt(2x+3)+sqrtx)$ $=$

$(x+3)/(sqrt(2x+3)+sqrtx)$

As a result,

$lim_(xrarr+oo)(sqrt(2x+3)-sqrtx)=lim_(xrarr+oo)(x+3)/(sqrt(2x+3)+sqrtx)$ $=$

$lim_(xrarr+oo)(x+3)/(sqrt(x^2(2/x+3/x^2))+sqrtx)$ $=$

$lim_(xrarr+oo)(x+3)/(|x|sqrt(2/x+3/x^2)+|x|sqrt(1/x))$ $=$

$x->+oo$
$x>0$

$lim_(xrarr+oo)(x+3)/(xsqrt(2/x+3/x^2)+xsqrt(1/x))$ $=$

$lim_(xrarr+oo)(cancel(x)(1+3/x))/(cancel(x)(sqrt(2/x+3/x^2)+sqrt(1/x))$ $=$

$lim_(xrarr+oo)(1+3/x)/(sqrt(2/x+3/x^2)+sqrt(1/x))=^((1/0^+)$ $+oo$

How do you find lim sqrt(2x+3)-sqrtxlim sqrt(2x+3)-sqrtx as x->oox->oo?