Calculus: Epsilon delta proof ?
How necessary is it to fully comprehend and perform the epsilon delta theorem on a limit in order to be competent in calculus?
I understand the basic concept of the theorem from a geometric standpoint, I can calculate the limit of most basic limits, one-sided limits, and infinite limits, I can take derivatives and calculate integrals, but when I look at the formal definition of a limit in the epsilon delta form my eyes glaze over and my brain goes numb.
When does this theorem explicitly come into play when doing general calculus, and do I really need to understand this proof?
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Generally, you won't need to know this proof to compute limits, or the idea of limits.
This epsilon and delta business will return in multi-variable limits, but again, it isn't used to actually compute the limits.
You'll probably have to use the proof to do 1-5 actual problems, but in general it's not useful in actually calculating limits. That being said, you should understand it fully, enough to actually employ it on demand. There's all kinds of tutorials and helpful stuff on the web. Go hunting and I'm sure you'll be able to not glaze over it.
Now, if you go beyond calculus, into the more abstract mathematics such as analysis, the general limit definition comes into play a lot more than the applied one you see in calculus, but the geometric interpretation is the same in most cases.