高等数学极限,跪求啊 为什么lim当x趋向于1,x⼀(x-1)=无穷 为什么lim当x趋向于1,2⼀(x^2-1)=无穷

2025-03-16 12:41:54
推荐回答(4个)
回答1:

第一题x/(x-1)=(x-1+1)/(x-1)=1+ 1/(x-1) ,所以你懂的
第二题2/(x^2-1)=1/(x-1) -1/(x+1),而1/(x-1) 由上题可知x→∞,lim1/(x+1)=1/2
无穷大量与常数的和,还是无穷大量

回答2:

1、X\X-1=X-1+1\X-1=1+1\X-1,所以当X趋近于1时,极限值是无穷,1+无穷=无穷
2、2\X平方-1=2\(X+1)(X-1) 分解因式得=(X+1)-(X-1)\(X+1)(X-1)=1\X-1 - 1\X+1,前面的式子的极限是无穷 第二个式子的极限是1\2,所以结果是无穷。
及时回复哦,谢谢!

回答3:

说明:求极限如果代入后分母是零,肯定是不能直接代入求的,一般分子分母对消一部分,或等价替换等一系列方法。

这2道题要用倒数法:由无穷大和无穷小的关系求极限。

第1题:
lim(x→1) x/(x-1)
=lim(x→1) 1/(x-1)
=∞

因为lim(x→1) (x-1)=0,也就是分母趋向于无穷小,倒过来的结果当然是无穷大。

根据高等数学极限定义:函数极限为无穷大时,认为极限不存在,这里暂时表述为极限是无穷大。

第2题:
lim(x→1) 2/(x²-1)=∞

同样的道理:因为lim(x→1)(x²-1)=0,也就是说分母趋向于无穷小(分母取不到0,是无限接近0,是一个无穷小),倒过来的结果当然是无穷大。

回答4:

lim x/(x-1) =lim 1/[1-(1/x)] 当x->1时,1-1/x ->0 1/[1-1/x]-->+∞
x->1 x->1

同样,下面那个 当x->1时 x^2 ->1 x^2-1->0 2/(x^2-1)->+∞

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