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等价无穷小替换的本质是 乘了一个极限为1的量，所以，才不会改变原极限。 从这里可以看到，是整体乘的，而不能在其中的某一个部分（加减法）去乘。 等价无穷小在一定条件下可以局部加减替换，具体规则为：加减项中如果每一项都是无穷小，且各自用等价无穷小替换后结果不为0，则可替换。 实际上，等价无穷小替换就是取的泰勒展开的主导项。 使用等价无穷小替换后主导项不被抵消。 三、为什么乘除时可以无顾忌地使用等价无穷小量替换？ 因为乘除不会消去主导项。 文章浏览阅读1.3k次，点赞7次，收藏6次。 求极限时什么时候可以在加减式中使用等价无穷小替换？ _加减法无穷小替换原则.
，一个视频带你无痛记忆所有等价无穷小替换，等价无穷小替换，加减法替换原则，两条概括，【数一147】【顶级理解系列】一道题搞定极限为啥有的能等价有的不能等价，来听听元哥的顶级理解，【高等数学】为什么等价无穷小不要轻易用在加减法？ 为什么有些题目在加减时替换能得到正确答案，有些则不能，它什么时候是可以用于加减的？ 其实产生这些疑问，都是因为并没有真正理解等价无穷小量的替换是个什么东西。 该题考察等价无穷小在加减运算中的替换条件。 解析要点如下： 1.基本替换原则：在乘除运算中，等价无穷小（如$x→0$时$sinx∼x$）可直接替换。 但在加减运算中需满足特殊条件。
（1） 等价无穷小替换只能在乘除式子中使用，而不能在加减式子中使用。
等价无穷小是一种极限的等价关系，它可以用来简化极限运算和证明极限的存在性。 二等价无穷小替换的条件1被代换的量，在取极限的时候极限值为0。 2被代换的量，作为被乘或者被除的元素时可以用等价无穷小代换，但是作为加减的元素时就不可以。 由此, 从泰勒的角度我们理解了为什么在乘法时, 等价无穷小可以直接做替换, 本质上是因为 高阶项 o(x), 在求极限时可以被忽略, 如果，有老师在讲等价无穷小替换时，提到“只要精度足够，就算加减中的项也是可以替换的”，却不讲清楚其本质是泰勒展开，那我认为，这是不负责任的，因为遇到复杂一点的极限，我们是很难判断精度这个问题的。
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