How do you find $lim root3(t+4)-root3t$ as $t->oo$ ?

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Answer 1 · 2017-03-01T18:35:44

$0$

$root3(t+4)-root3t = root3(t)(root3(1+4/t)-1)$

but

$root3(1+4/t) = 1+1/3(4/t)+(1/3(1/3-1))/(2!)(4/t)^2+cdots$

and

$root3(t)(root3(1+4/t)-1) = root3(t)(-1/3(4/t)-(1/3(1/3-1))/(2!)(4/t)^2-cdots) =$

$=1/t^(2/3)(-4/3+f(1/t))$ so

$lim_(t->oo)root3(t+4)-root3t = lim_(t->oo)1/t^(2/3)(-4/3+f(1/t))=0$

How do you find lim root3(t+4)-root3tlim root3(t+4)-root3t as t->oot->oo?