(수정 중)
precise limits of function as x approaches a constant
The following problems require the use of the precise ε,δ definition of limits of functions as x approaches a constant. Most problems are average.
We will begin with the precise ε,δ definition of the limit of a function as x approaches a constant.
DEFINITION: The statement  has the following precise definition. Given any real number ε>0, there exitst another real number δ>0 so that
if  , \then
In general, the value of δ will depend on the value of ε. That is, we will always begin with ε>0 and then determine an appropriate corresponding value for δ>0. There are many values of δ which work. Once you find a value that work, all smaller values of δ also work.
To try and understand the meaning behind this abstract definition, see the given diagram below.

We first pick an ε band around the number L on the y-axis. We then determine a δ band around the nuber a on the x-axis so that for all x-values (excluding x=a) inside the δ band, the corresponding y-values lie inside the ε band. In other words, we first pick a prescribed closeness (ε) to L. Then we get close enough (δ) to a so that all the corresponding y-values fall inside the ε band. If a δ>0 can be found for each value of ε>0, then we have proven that L is the correct limit. If there is a singel ε>0 for which this process fails, then the limit L has been incorrectly computed, or the limit does not exist.

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는 를 포함하는 어떤 개구간(는 제외될 수 있음)에서 정의된 함수라고 하자.

두 입체도형을 일정한 평면에 평행한 평면으로 잘랐을 때, 두 단면의 면적의 비가 언제나 m:n이면 두 입체도형의 부피 비는 m:n이다.  일정한 평면에 수직한 한 직선을 축이라 한다. 

Q) 이 자연수일 때,  이 소수이면,  도 소수임을 증명하라.