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When I read the book, I came across the term "higher order infinitesimal". I know what "infinitesimal" means, but what is a higher order infinitesimal?

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Just for review:

The order of ___ is the largest exponent of the independent variable. (e.g. $f(x)=x^3+x^2$'s order is $3$.

The order of an infinitesimal number is the "number of infinities" it has.

Some examples:

$$x=\frac1\infty$$ The order of infinitesimal number, $x$, is $1$. $$x=\frac{1}{\infty^2}$$ The order of infinitesimal number, $x$, is $2$. $$x=\frac{10\cdot\sqrt{\infty}}{\infty^2}$$ The order of infinitesimal number, $x$, is $1.5$.

Infinitesimal numbers are usually used in limits, because infinitesimal numbers always limit to 0.

Hopefully this makes sense!

The order of an infinitesimal number is typically used in calculating limits that are indeterminate, such as: $$\lim_{x\to1}\frac{x-1}{x^2-x}=\lim_{x\to1}\frac{1}{x-1}$$ By identifying the denominator has a higher order, it can be determined that the fraction is a first order infinitesimal when $x$ nears $1$.

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    This is full of mathematical errors, not the least of which is writing $1/\infty$ and expecting it to mean something. – 2017-01-14
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    $\infty$ is an "absorbing" infinite number; e.g. $\infty$ and $\infty^2$ are the same thing. There are notions of infinite numbers $H$ such that $H$ is different from $H^2$, but they have no (direct) relationship with $\infty$. – 2017-10-04