(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.
는 를 포함하는 어떤 개구간(는 제외될 수 있음)에서 정의된 함수라고 하자.
만약 임의의 양수 에 대하여 일 때마다 을 만족하는 이 존재하면, 가 에 가까이 접근할 때 의 극한이 이라 정의한다.
