Techniques of differentiation

Understand limits and derivatives · Diagrams

Distinguish a missing value from a missing limit

A missing value can still have a limit

Use radians. The curve sin(x)/x approaches 1 from both sides, but no function value is assigned at x = 0. The open circle stays open.

sin(x)/x, x ≠ 0 · radians

Distance to the limit point

More precision

View

Compare the approach
Position	x	f(x)
Left	-0.1	0.998334166
At point	0	Undefined
Right	0.1	0.998334166

At distance 0.1, compare both sides. Limit = 1. f(0) remains undefined.

Values are rounded to 9 decimal places. Use the table to inspect points that overlap on the graph.

Near zero zooms the x-axis to separate the sample points from the hole. In the whole view, overlapping sample markers are hidden; the table keeps their values.

Nearby values illustrate the limit; a finite table alone is not a proof.

Teaching and source notes

The source graph has an open circle at (0, 1). No value is assigned to sin(x)/x at x = 0, and the explorer keeps a gap in the curve there. The two-sided limit is 1 when x is measured in radians.

The approach table supports comparison; it is not a proof. Establishing this standard limit requires an argument such as the squeeze theorem. A rounded value of 1 does not remove the domain hole.

The graph of sin(x)/x on [−3π,3π] has an open circle at (0,1). Its limit is 1 as x approaches 0, despite being undefined at 0. Angles are in radians.

Open original diagram

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