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College simple high number limit
Abandon this equivalent infinitesimal algorithm! The concept of "equivalent infinitesimal" itself is somewhat vague. The reason why addition and subtraction can't become equivalent infinitesimal is because "equivalent infinitesimal" will bring a problem. Two factors are equivalent infinitesimal of several orders. Tanx and sin x in x->; 0 is an equivalent infinitesimal, but the order of two infinitesimals is different, or the approximation speed of infinitesimals is different.

The safest algorithm is Taylor expansion: for tanx and sinx Taylor expansion in x- > at 0, yes.

tanx = x + ( 1/3)x^3 +o(x^5)

sinx = x - ( 1/6)x^3 + o(x^5)

It is enough to expand to X 3 here, because the denominator is only X 3. After the expansion is too high, those tail terms become high-order infinitesimals.

So tanx-sinx = (1/2) x 3+o (x 5)

So (tanx-sinx) = (1/2)+[o (x 5)]/x 3, and the result of taking the limit is1/2.

Through Taylor expansion, it is found that tanx and sinx are in x->; 0 is indeed an infinitesimal, but in order x 3, the approximation speeds of the two are different, so the two infinitesimals are not completely equivalent. But when we only consider the case of order X, it is generally called "equivalent infinitesimal".