Question

How do you prove limit of `2/sqrt(x-4)=oo` as `x->4^+` using the precise definition of a limit?

Answer 1

Given any number `M > 0` choose `delta_M < 4/M^2`.

Then for `x in(4, 4+delta_M)` we have that `x - 4 < delta_M` and so:

`2/sqrt(x-4) > 2/sqrt(delta_M) > 2/sqrt(4/M^2) = M`

We can then conclude that for any `M` we can choose a `delta_M` such that:

`x in(4, 4+delta_M) => 2/sqrt(x-4) > M`

which proves the limit.