Question

My proof for this limit using the definition is correct? `lim to 2^+ (1/(x-2)) = +\infty`

My answer:

For all A > 0, exists `\delta` > 0 such that:
`(1/(x-2)) > A` so that 0 < x+2 < `\delta`.

Looking on inequality bellow between B, we have the key choose for `\delta` :

`(1/(x-2)) > A`
`(x-2) < 1/A`
`x < 1/A+2`

Like this, for `\delta` = `1/A+2`, we have `1/(x-2) > A` always that 0 < x-2 < `delta`.

Answer 1

See explanation

Explanation:

There is one mistake: `0 < x+2 < delta`. After all, it is `x-2` that you want to go towards 0.

I might also want to refine the wording a little, for instance:
"For all A > 0, there exists a `delta>0` such that:" to make the proof clearer.

Also, as the proof presupposes that `x>2`, I might write:
`2<x<2+1/A` to make it clear that x lies between 2 and `2+1/A`.

One other detail: You introduce B, but it's not clear where B belongs or what it refers to.