Algebra of Limits

Evaluating a limit directly from a table of values, as in Section 1, is

instructive but impractical for anything beyond the simplest functions. The algebra of limits is the set of rules that lets the limit of a combination of functions -- a sum,

difference, product, or quotient -- be built up from the limits of the simpler functions that

make it up, without returning to tables or first-principles reasoning every time.

Two starting facts. For any real constant $c$ and any real $a$,

since a constant function never changes value, and $f(x)=x$ trivially approaches $a$ as $x$

approaches $a$. Every other limit rule in this chapter is built out of these two facts together

with the theorem below.

Theorem (algebra of limits). Suppose $\displaystyle\lim_{x\to a}f(x) = l$ and

$\displaystyle\lim_{x\to a}g(x) = m$, where $l$ and $m$ are both finite real numbers. Then:

1. Sum rule: $\displaystyle\lim_{x\to a}\big[f(x)+g(x)\big] = l+m$.
2. Difference rule: $\displaystyle\lim_{x\to a}\big[f(x)-g(x)\big] = l-m$.
3. Product rule: $\displaystyle\lim_{x\to a}\big[f(x)\cdot g(x)\big] = l\cdot m$.
4. Quotient rule: $\displaystyle\lim_{x\to a}\frac{f(x)}{g(x)} = \frac{l}{m}$, provided $m\neq0$.
5. Scalar multiple rule: $\displaystyle\lim_{x\to a}\big[k\cdot f(x)\big] = k\cdot l$, for any constant $k$ -- a special case of the product rule with $g(x)=k$.

These five rules are stated here as the working toolkit of the chapter (a fully rigorous proof

needs the formal definition of a limit, beyond this chapter's intuitive treatment), but each is

entirely natural: if $f(x)$ is settling down near $l$ and $g(x)$ near $m$, then $f(x)+g(x)$

should settle down near $l+m$, and so on.

Why the quotient rule needs $m\neq0$. If $m=0$, the combination $f(x)/g(x)$ can behave in

several different ways as $x\to a$ -- it can grow without bound, oscillate, or (if $f$ also

tends to $0$) settle down to a genuine finite limit reached only after cancelling a common

factor. This last case, where BOTH $l=0$ and $m=0$ simultaneously, is called an indeterminate form of type $0/0$, and it is the single most important case in the rest of this chapter --

the quotient rule above simply does not apply, and a different technique (factoring, in Section

3; a standard limit, in Sections 4 and 5) is needed instead. …