Nothing new has to be proved. Every rule about combining sequence limits becomes a rule about combining function limits, because the sequential characterisation turns a statement about a function into a statement about arbitrary sequences of inputs. The same list comes across: sums, differences, products, constants, quotients with a non-zero denominator, order and the squeeze.

A currency conversion that preserves sums and products lets every arithmetic identity be checked in either currency. The characterisation is that conversion, and after it is established the rules do not have to be re-derived — they are transported.

Suppose $\lim_{x\to c}f(x)=L$ and $\lim_{x\to c}g(x)=M$. Then $f\pm g\to L\pm M$, $fg\to LM$, $kf\to kL$ for a constant $k$, and $\tfrac fg\to\tfrac LM$ provided $M\ne0$. If $f\le g$ near $c$ then $L\le M$, and if $f\le h\le g$ near $c$ with $L=M$ then $h\to L$ as well.

• Each rule requires both limits to exist first. $\lim(f+g)$ can exist when neither $\lim f$ nor $\lim g$ does.
• The quotient rule needs $M\ne0$. When and the quotient has no finite limit; when both are zero anything may happen, and that case is the whole of the seventh chapter.

Transport the statement rather than proving it again. Take an arbitrary sequence of inputs approaching the point and avoiding it. The characterisation turns each hypothesis into a statement about the sequence of values, the product rule for sequences combines them, and the characterisation read the other way turns the conclusion back into a statement about the function. Nothing about products is used except the rule already proved in the fourth chapter.

Take an arbitrary sequence of inputs of the kind the characterisation speaks about.

The characterisation applied to each hypothesis.

The product rule for sequences, proved in chapter four.

The sequence was arbitrary, so the characterisation read backwards gives the function limit.

The rules combine limits that already exist. Read backwards they are false.

Trap the awkward function between two that agree in the limit.

• Sums, differences, products, constants and quotients behave as expected, given that the separate limits exist.
• The quotient rule needs the denominator limit to be non-zero.
• Order passes weakly and the squeeze theorem holds.
• A polynomial has its value as its limit everywhere.

Great work! You now know how to:

• Apply the rules for sums, products, constants and quotients to function limits
• Name the hypothesis each rule needs
• Use the squeeze theorem on a bounded oscillation
• Say why the rules needed no new proof