Techniques of differentiation

Understand limits and derivatives · Diagrams

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.

Open original diagram

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