[分析] 1st and 2nd order approximation thm

[cotton757 (cotton757)]

Calaulus the following limits by applying the 1st and 2nd order approximation.

               sin(x+xy-y)-(x+y)
    lim     -------------------------
(x,y)->(0,0)    (x^2+y^2)^(1/2)

                 e^(x-y)-1-x+y
    lim     -------------------------
(x,y)->(0,0)       x^2 + y^2

以上兩題極限皆不存在

所以代表不能用下面的兩個定理

我想請問什麼會這樣呢

請板上高手解釋一下吧

免得考試時換個函數型態我就不會寫了

thm. Let A be an open subaet of Rn and suppose the function is continuous

     differentiable. Let x in A

          f(x+h)-[f(x)+<▽f(x),h>]
     lim ------------------------- = 0
    h->0         ||h||


thm. Let A be an open subaet of Rn and suppose the function is continuous

     second order differentiable. Let x in A

          f(x+h)-[f(x)+<▽f(x),h>+(1/2)<▽^2f(x)h,h>]
     lim --------------------------------------------- = 0
    h->0                     ||h||^2


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