Question

How do you prove the statement lim as x approaches 0 for `x^2 = 0` using the epsilon and delta definition?

Answer 1

See the explanation, below.

Explanation:

You need to show that if we are given a positive number that we'll call `epsilon`, then there is a number `delta` (also positive) that makes the following true:

if `x` is chosen so the `0 < abs(x-0) < delta`, then `abs(x^2-0) < epsilon`.

One way to do this is to notice that if `absx < sqrt(epsilon)`, the `absx < epsilon`

Choose `delta = sqrt epsilon`

Another way is to observe that for `absx < 1`, we have `abs(x^2) < absx`. Choose `delta = min {1,epsilon}`

Whichever way is chosen, we then write up the proof:

Proof

Given `epsilon > 0`, choose `delta = "whatever"`

Now if `0 < abs(x-0) < delta`, then

[insert a proof that `absx < 0` implies `abs(x^2) < epsilon`]