What is the limit of 1/(sqrt(x)) as it goes to infinity?

1 Answer
Feb 21, 2018

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

Explanation:

Intuitively we can see that as x->oo also sqrtx->oo because it is positive, increasing and not bounded and thus 1/sqrtx can be made as close to zero as we want.

More formally, given any epsilon > 0 we can choose M such that sqrtM> 1/epsilon and then, as sqrtx is an increasing function:

x > M => sqrtx > sqrtM => sqrtx > 1/epsilon => 1/sqrtx < epsilon

so for every epsilon > 0 we can find M > 0 such that:

x > M => 0 < sqrtx < epsilon

which demonstrates the limit.