Question

大学高数极限问题

大学高数极限问题4 5 6小题 4小题做到图上那一步就做不下去了，56小题完全下不了笔

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Answer 1

如图

追问

4小问答案是-2也 5小问怎么化的啊为什么x都没了

追答

4是2

x用2代入

追问

4分母的-xy是怎么化的啊

追答

Answer 2

(4)
lim((x,y)->(0,0)) xy/[√(2-e^(xy)) -1 ]
=lim((x,y)->(0,0)) xy.[√(2-e^(xy)) +1 ]/ [ 1 - e^(xy) ]
=lim((x,y)->(0,0)) xy.[√(2-e^(xy)) +1 ]/ (-xy)
=lim((x,y)->(0,0)) -[√(2-e^(xy)) +1 ]
= -(1+1)
=-2
(5)
lim((x,y)->(2,0)) sin(xy)/y
=lim((x,y)->(2,0)) xy/y
=lim((x,y)->(2,0)) x
=2
(6)
lim((x,y)->(0,0)) [ 1- cos(x^2+y^2) ]/[ (x^2+y^2).e^(x^2.y^2) ]
=lim((x,y)->(0,0)) (1/2)(x^2+y^2)^2 /[ (x^2+y^2).e^(x^2.y^2) ]
=lim((x,y)->(0,0)) (1/2)(x^2+y^2) /e^(x^2.y^2)
=0

追问

4小问为什么分母直接就变成xy了

追答

(x,y)->(0,0)
e^(xy) ~ 1+xy
1- e^(xy) ~ -xy
(4)
lim((x,y)->(0,0)) xy/[√(2-e^(xy)) -1 ]
=lim((x,y)->(0,0)) xy.[√(2-e^(xy)) +1 ]/ [ 1 - e^(xy) ]
=lim((x,y)->(0,0)) xy.[√(2-e^(xy)) +1 ]/ (-xy)
=lim((x,y)->(0,0)) -[√(2-e^(xy)) +1 ]
= -(1+1)
=-2

追问

e^（xy）为什么等于1+xy呀

追答

e^x ~ 1+x
等价无穷小
e^(xy) ~ 1+xy