Question

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

Answer 1

`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.