Equivalent functions

From the standard limits one obtains a list of handy approximations. We use the symbol $\square$ to denote any quantity that tends to zero (it may be $x$, or $x^2$, or $\sin x$, or any infinitesimal expression).

In summary — List of equivalent functions

When $\square\to 0$:

\sin(\square) &\sim \square \\ \cos(\square) &\sim 1 - \frac{\square^2}{2} \quad\text{(e quindi } 1-\cos(\square)\sim \square^2/2\text{)} \\ \tan(\square) &\sim \square \\ e^{\square} &\sim 1 + \square \\ (1+\square)^k &\sim 1 + k\,\square \quad\text{(anche per } k \text{ frazionario o negativo)} \\ \ln(1+\square) &\sim \square \end{aligned}$$ **Inside $\square$ one may place any quantity that tends to zero.** For example, if $x\to +\infty$ one may set $\square = 1/x$ (which tends to $0$) and write $\sin(1/x)\sim 1/x$. The arguments of $\sin$ and $\cos$ must be in radians.

Remark — When substitution is allowed and when it is not

These substitutions are legitimate “inside” a limit provided the substituted function appears in the numerator or denominator as a factor, not as a term added to other terms of the same order. In the latter case the second-order approximation is needed, for example $e^\square \sim 1+\square+\square^2/2$.

Links

Topics: Limits
Concepts: Equivalent functions · Standard limits
Skills: Using formulae