STEVIE-G™ asked in 科學及數學數學 · 1 decade ago

Definition of Limit

Update:

我一開始由|x - c| < ε3, |f(x) - L| < δ/2 及 |x - c| < ε4, |g(x) - M| < δ/2 做起有冇問題,好似http://hk.knowledge.yahoo.com/question/question?qi...

1 Answer

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  • 1 decade ago
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    By the definition of limit, we have:

    For a sufficiently small positive values ε1 and ε2, we can say that:

    When |x - c| < ε1, |f(x) - L| < δ

    AND

    When |x - c| < ε2, |g(x) - M| < δ

    where δ is also a sufficiently small positive value.

    Now, for f(x) - g(x), we can find ε3 < ε1 and ε4 < ε2 such that:

    When |x - c| < ε3, |f(x) - L| < δ/2

    AND

    When |x - c| < ε4, |g(x) - M| < δ/2

    We now have, when |x - c| < min(ε3, ε4):

    |[f(x) - L] - [g(x) - M]| <= |f(x) - L| + |g(x) - M|

    |[f(x) - g(x)] - (L - M)| <= δ

    Hence lim (x → c) [f(x) - g(x)] = L - M

    2009-12-10 20:58:19 補充:

    沒問題, 因為 ε1, ε2, ε3 和 ε4 都只說明是 very small, 並沒有一個 definite 的 value.

    Source(s): Myself
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