As far as I know, we use the ratio limit of two infinitesimals to reflect the speed of two infinitesimals approaching zero. When their ratio limit is equal to - 1, is it equivalent? For example, when x → 0, X and - X are infinitesimals, and the limit of their ratio is equal to - 1, so they tend to zero at the same speed. Theoretically, they are equivalent, but the equivalent definition of Infinitesimal in advanced mathematics textbooks is that the ratio limit is equal to 1, not the absolute value of the ratio limit is equal to 1

As far as I know, we use the ratio limit of two infinitesimals to reflect the speed of two infinitesimals approaching zero. When their ratio limit is equal to - 1, is it equivalent? For example, when x → 0, X and - X are infinitesimals, and the limit of their ratio is equal to - 1, so they tend to zero at the same speed. Theoretically, they are equivalent, but the equivalent definition of Infinitesimal in advanced mathematics textbooks is that the ratio limit is equal to 1, not the absolute value of the ratio limit is equal to 1

In fact, strictly speaking, the definition of Equivalent Infinitesimal in advanced mathematics textbooks is more accurate. For example, in your example, when x → 0, X and - X are infinitesimal, and the limit of their ratio is equal to - 1, so they tend to zero at the same rate, but they tend to zero from different directions. Strictly speaking, the rate and direction of approaching zero should be the same