Equivalent functions: a tool for calculating limits

When two functions have the same finite, nonzero limit at a point, possibly at infinity, their quotient tends to 1. This a priori trivial observation becomes useful when both limits are either zero or infinite. One function can then replace the other when finding the limit of a product or quotient. The two functions are said to be equivalent in a neighborhood of that point. In fact, this definition lacks rigor, so a more precise one is generally given (see box). Whenever an expansion is available, its first nonzero term is equivalent to the function near the point under study. From now on, we shall assume that this point is 0, translating the variable if necessary.

First examples -----------------

The third-order Taylor expansion of the sine function at 0 is

$\sin x=x-{\displaystyle \frac{x^3}{6}}+o\left(x^3\right),\backslash sin\; x\; =\; x-\; \backslash dfrac\{x^3\}\{6\}\; +\; o\; \backslash left(x^3\backslash right),$ so the sine function is equivalent to the function x ↦ x near 0, since ${\displaystyle \frac{\sin x}{x}}=1-{\displaystyle \frac{x^2}{6}}+o\left(x^2\right),\backslash dfrac\{\backslash sin\; x\}\{x\}\; =\; 1-\; \backslash dfrac\{x^2\}\{6\}+\; o\; \backslash left(x^2\backslash right),$ which clearly tends to 1 as x tends to 0.

Now let h be the product of two functions f and g, so that h(x) = f (x) g (x), and suppose that g is equivalent to g₁. Then $h(x)=f(x)g_1(x){\displaystyle \frac{g(x)}{g_1(x)}},h(x)\; =\; f(x)g\_1(x)\; \backslash dfrac\{g(x)\}\{g\_1(x)\},$ provided that g₁(x) is nonzero except possibly at 0. It follows that h has a limit at 0 if and only if h₁(x) = f (x) g₁(x) does, and in that case the two limits are equal because ${\displaystyle \frac{g(x)}{g_1(x)}}\backslash dfrac\{g(x)\}\{g\_1(x)\}$ tends to 1.