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

1 Answer
Jim H
Oct 9, 2015

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]

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