The definition is the one from the fourth chapter with the index replaced by a distance. A tolerance on the output is given first; a distance on the input must then be produced, such that every input within that distance of the point, other than the point itself, has its value within the tolerance. Where a sequence proof chose a place, a function proof chooses a radius, and everything else about the two is the same.

A machine must deliver a rod within a hundredth of a millimetre of the specified length. Your job is to say how accurately the dial must be set. A tighter specification on the output demands a tighter setting on the input, and the promise is exactly that such a setting always exists.

The tolerance $\varepsilon$ is given, drawing the band round $L$ in the values. The radius $\delta$ is then chosen so that everything within it of $c$, except $c$ itself, is sent inside that band. A smaller $\varepsilon$ usually forces a smaller $\delta$.

$\lim_{x\to c}f(x)=L$ means: for every $\varepsilon>0$ there is a $$ such that for every in the domain with we have . The strict inequality is what excludes the point itself. A proof has the four lines of an epsilon–N proof with replaced by , and the scratch work is again solving an inequality backwards.

• $\delta$ is chosen after $\varepsilon$ and may depend on it and on $c$. A single $\delta$ answering every $\varepsilon$ would force $f$ to be constant near .

Simplify the distance between the value and the candidate first: for a linear function it is a constant multiple of the distance between the input and the point. The tolerance then translates directly into a radius by dividing, and the proof is written forwards from that choice. Nothing in the argument depends on the particular numbers, which is why every linear function is handled the same way.

The tolerance comes first and the radius will be chosen to suit it.

Simplify the distance between the value and the candidate limit.

Divide the tolerance by the factor the simplification produced.

Take any input within that radius of the point and different from it.

Which is the definition, for the tolerance given — and it was arbitrary.

The quantifiers are in the same order as in the sequence definition, with the place replaced by a radius.

For anything nonlinear, cap the radius at one before estimating, so the awkward factor can be bounded on a fixed interval.

• The tolerance comes first and the radius answers it.
• $0<\lvert x-c\rvert$ excludes the point, which is what makes a limit a limit.
• Any working radius is acceptable, and smaller ones work too.
• For anything nonlinear, cap the radius first and bound the awkward factor on that interval.

Great work! You now know how to:

• State the definition with its quantifiers in order
• Write an epsilon–delta proof for a linear function
• Cap the radius first when the function is nonlinear
• Negate the definition correctly