$lim_(t->oo)(sqrt(t)-1)/(t-1) =$ ?

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Answer 1 · 2016-09-20T00:46:47

I think it was for $t->oo$ ...giving zero in the limit.

If it is $t->oo$ :

$lim_(t->oo)(sqrt(t)-1)/(t-1)=oo/oo$

To solve this we can use de L'Hospital Rule deriving top and bottom to get:
$lim_(t->oo)(1/(2sqrt(t)))/1=lim_(t->oo)1/(2sqrt(t))$

as $t->oo$ the argument tends to zero;

$lim_(t->oo)1/(2sqrt(t))=0$

Answer 2 · 2016-09-23T07:45:32

$lim_(x->oo)(sqrt(t)-1)/(t-1)=0$

An alternative method, solving algebraically:

$lim_(x->oo)(sqrt(t)-1)/(t-1)$

$=lim_(x->oo)((sqrt(t)-1)(sqrt(t)+1))/((t-1)(sqrt(t)+1))$

$=lim_(x->oo)(t-1)/((t-1)(sqrt(t)+1))$

$=lim_(x->oo)1/(sqrt(t)+1)$

$=1/oo$

$=0$

Answer 3 · 2016-09-23T07:55:34

Making $y = sqrt t$

$lim_(t->oo)(sqrt(t)-1)/(t-1) equiv lim_(y->oo)(y-1)/(y^2-1) = lim_(y->oo)1/(y+1) = 0$

lim_(t->oo)(sqrt(t)-1)/(t-1) = lim_(t->oo)(sqrt(t)-1)/(t-1) = ?