Techniques of differentiation

Understand limits and derivatives · Diagrams · Complete reading

Read the same limit from a curve and a table

Approach the same limit from both sides

For f(x) = x² − x + 2, move towards x = 2 and compare the curve with the table. Here f(2) = 4 as well as the limit being 4.

x² − x + 2

Distance to the limit point

More precision

Compare the approach
Position	x	f(x)
Left	1.9	3.71
At point	2	4
Right	2.1	4.31

At distance 0.1, compare both sides. Limit = 4. f(2) = 4.

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

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

Teaching and source notes

The retained source uses f(x) = x² − x + 2 and approaches x = 2 from both sides. Substitution gives f(2) = 4. Polynomial continuity justifies this limit; a small table alone is not its proof.

The approach control never substitutes h = 0. Rounded nearby function values need not equal 4 exactly, even when the display rounds them to 4. The linked original retains the printed guide projections.

For y=x²−x+2, both sides of x=2 approach y=4. Dashed projections identify x=2 and f(x)=4.

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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.

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Let a secant approach a tangent

Move a secant towards the tangent

Illustrative example: y = x² with P = (1, 1). Move Q = (1 + h, (1 + h)²) towards P. The source sketch uses a general smooth curve, not this particular formula.

Illustrative curve y = x²

Size of h

More precision

Approach from

Points on the curve
Point	x	f(x)
P · fixed	1	1
Q · moving	1.1	1.21

The dashed line is the tangent at P; the segment PQ is the secant.

h = 0.1. Secant gradient = 2.1; tangent gradient = 2.

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

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

Teaching and source notes

The source sketch is a general smooth curve with P(x, y), Q(x + h, y + Δy), secant PQ and tangent AB. This numerical example is explicitly y = x² at P = (1, 1), not a recovered formula for the original drawing.

For h ≠ 0, Δy = (1 + h)² − 1 = h(2 + h), so Δy/h = 2 + h. Its limit is 2 from either side. At h = 0 the difference quotient is undefined; the derivative comes from its limit. A secant need not tend to a tangent where the derivative does not exist.

An illustrative smooth curve shows P(x,y), Q(x+h,y+Δy), secant PQ and tangent AB. The difference quotient Δy/h tends to the tangent gradient as h approaches 0 if the derivative exists.

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