點証明　L ' Hosital Rule ?

Proofs of l'Hpital's rule

Proof by Cauchy's mean value theorem
The most common proof of l'Hpital's rule uses Cauchy's mean value theorem.

With the indeterminate form 0 over 0
The case when
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First, we expand continuously (or redefine) f(x) and g(x) by 0 for x = c. This doesn't change the limit since the limit doesn't depend on the value in the point (by definition).
According to Cauchy's mean value theorem there is a constant ξ in c < ξ < c + h such that:


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Since f(c) = g(c) = 0,


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If
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and


圖片參考：http://upload.wikimedia.org/math/7/e/7/7e7d16c2e5858ebb78b46c4c9d44bdb7.png

With the indeterminate form infinity over infinity
The case when
圖片參考：http://upload.wikimedia.org/math/2/8/a/28a6a86109cc550a54fa9ad4e3a07ba4.png

Let x < y < x + h. Then using Cauchy's mean value theorem:


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We rewrite that in the form


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and then by the discussion of the two cases


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we show that the limit of f(x)/g(x) tends to the same when
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.