Limits and continuity, explained one step at a time

Limits look abstract until you see them as a question about behaviour near a point. Noeta walks through the algebraic manipulation that removes the indeterminate form, and keeps testing the case you get wrong.

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What this covers

One-sided and two-sided limits
Indeterminate forms 0/0 and ∞/∞
Factoring and rationalising techniques
Limits at infinity and asymptotes
Continuity and the squeeze theorem

Worked example

Problem

Evaluate lim(x→3) (x² − 9)/(x − 3)

1
Substitute first
Direct substitution gives 0/0, an indeterminate form, so the expression must simplify.
2
Factor
x² − 9 = (x − 3)(x + 3).
3
Cancel
The (x − 3) factors cancel, leaving x + 3 for all x ≠ 3.
4
Substitute again
3 + 3 = 6.

Answer: 6

Mistakes that cost marks

Concluding the limit does not exist as soon as substitution gives 0/0.
Cancelling a factor and forgetting the original function is still undefined at that point.
Applying L'Hôpital's rule to a form that is not indeterminate.

Questions students ask

Do I need limits before derivatives?

Yes — the derivative is defined as a limit, and shaky limit work shows up later as chain-rule confusion.

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