I understand I can show the function $e ^ x$ is continuous at point $0$, by means of the definition epsilon delta. But which epsilon I should take?

Question

I understand I can show the function $e ^ x$ is continuous at point $0$, by means of the definition epsilon delta. But which epsilon I should take?

This is how far I've got: I think I should analyze when $x> 0$ and when $x <0$, but I'm not sure. I got two deltas: $\ln (\varepsilon + 1)$ and the other $\ln \left(\dfrac{1}{1 + \varepsilon}\right)$

I'm confused I do not know which delta I should choose to try this

Answer 1

One of those is negative. A negative $\delta$ is never appropriate, because there are absolute values in everything. We can't have an absolute value be less than something negative.

The larger issue - how can we talk about the logarithm when we don't even know that the exponential is continuous yet? Anything based on the logarithm here is likely to be invalid due to circular reasoning.

This calls for the fundamentals of how you're defining the exponential, to avoid that circularity. There are several reasonable choices, and we can't proceed further until we know what you're using.