Examples with equivalent functions

Example — First-order substitution

$\lim_{x\to 0}\frac{\sin(3x)}{x} = \lim_{x\to 0}\frac{3x}{x} = 3.$ Here $\sin(3x)\sim 3x$ (with $\square = 3x\to 0$), a legitimate substitution because $\sin(3x)$ is a factor.

Example — When the second order is needed

$\lim_{x\to 0}\frac{e^x - 1 - x}{x^2}.$ The approximation $e^x-1\approx x$ is not enough: it would give $0/0$, because $e^x-1-x$ is a sum in which the first-order term cancels. The second order is needed: $e^x \approx 1+x+\dfrac{x^2}{2}$, whence $e^x-1-x\approx \dfrac{x^2}{2}$, and the limit is $\dfrac{1}{2}$.

Comparing the two examples reveals the practical rule: substitute at first order when the function is an isolated factor; go up to second order when the first-order terms cancel out in a sum.

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Topics: Limits
Concepts: Indeterminate forms · Equivalent functions
Skills: Calculating limits