题目
求多元函数的极限
(x^2+y^2)e^(y-x) 其中x→+∞,y→-∞
(x^2+y^2)e^(y-x) 其中x→+∞,y→-∞
提问时间:2021-04-01
答案
∵lim(x->+∞,y->-∞)[(x-y)^2/e^(x-y)]
=lim(t->+∞)(t^2/e^t) (令t=x-y)
=lim(t->+∞)(2t/e^t) (∞/∞型极限,应用罗比达法则)
=lim(t->+∞)(2/e^t) (∞/∞型极限,应用罗比达法则)
=0
lim(x->+∞)(x/e^x)
=lim(x->+∞)(1/e^x) (∞/∞型极限,应用罗比达法则)
=0
lim(y->-∞)(ye^y)
=lim(y->-∞)[y/e^(-y)]
=lim(y->-∞)[-1/e^(-y)] (∞/∞型极限,应用罗比达法则)
=0
∴lim(x->+∞,y->-∞)[(x^2+y^2)e^(y-x)]
=lim(x->+∞,y->-∞)[((x-y)^2-2xy)/e^(x-y)]
=lim(x->+∞,y->-∞)[(x-y)^2/e^(x-y)-2(x/e^x)(ye^y)]
=lim(x->+∞,y->-∞)[(x-y)^2/e^(x-y)]-2*lim(x->+∞,y->-∞)(x/e^x)*lim(x->+∞,y->-∞)(ye^y)
=0-2*0*0
=0.
=lim(t->+∞)(t^2/e^t) (令t=x-y)
=lim(t->+∞)(2t/e^t) (∞/∞型极限,应用罗比达法则)
=lim(t->+∞)(2/e^t) (∞/∞型极限,应用罗比达法则)
=0
lim(x->+∞)(x/e^x)
=lim(x->+∞)(1/e^x) (∞/∞型极限,应用罗比达法则)
=0
lim(y->-∞)(ye^y)
=lim(y->-∞)[y/e^(-y)]
=lim(y->-∞)[-1/e^(-y)] (∞/∞型极限,应用罗比达法则)
=0
∴lim(x->+∞,y->-∞)[(x^2+y^2)e^(y-x)]
=lim(x->+∞,y->-∞)[((x-y)^2-2xy)/e^(x-y)]
=lim(x->+∞,y->-∞)[(x-y)^2/e^(x-y)-2(x/e^x)(ye^y)]
=lim(x->+∞,y->-∞)[(x-y)^2/e^(x-y)]-2*lim(x->+∞,y->-∞)(x/e^x)*lim(x->+∞,y->-∞)(ye^y)
=0-2*0*0
=0.
举一反三
我想写一篇关于奥巴马的演讲的文章,写哪一篇好呢?为什么好
奥巴马演讲不用看稿子.为什么中国领导演讲要看?
想找英语初三上学期的首字母填空练习……
英语翻译
1,人们染上烟瘾,最终因吸烟使自己丧命.
最新试题
- 1.-------与我同行作文
- 2水果店有苹果和梨一共250千克,已知苹果的1/5比里的1/3多2千克,水果店有苹果()千克?
- 3某整数,若加上12,则为正数,若加上10,则为负数,那么这个的平方为多少?
- 4Why do not you tell me the truth 改为同义句
- 5双曲线x2/4-y2/36=1上任一点到M到两渐近线的距离乘积值.把这结论推广到一般的双曲线
- 6关于城乡环境综合治理的文章
- 7You are one of the girls who _____(be) going to Beijing with me .填 is 还是 are 说理由.
- 8近似数4.2与4.20的大小相等,精确度也相同._.(判断对错)
- 9将同一重物举高,如图19所示,使用滑轮组的机械效率与使用斜面的机械效率之比为8:7,
- 10小小的花儿居然有如此的气魄!仿写一句
热门考点