Guessing a limit from a table is risky — floating-point cancellation can give wrong answers for very small increments. Algebraic computation using the limit laws is reliable. This note establishes the ten limit laws, introduces the Direct Substitution Property (the easiest case), and then handles the three situations where direct substitution fails: removable zeros, products with indeterminate form, and oscillating quotients. The squeeze theorem closes the section — it is the key tool for limits involving and near zero, and recurs in the proof of the derivative of .
Learning objectives. By the end of this note you should be able to:
- state all ten limit laws and identify which hypothesis each requires;
- evaluate any limit where direct substitution is valid;
- factorise, rationalise, or apply the squeeze theorem to evaluate limits that produce at first substitution;
- explain why depends only on near , not at .
In this lesson
- The limit laws
- Direct Substitution Property
- Algebraic techniques for indeterminate forms
- The Squeeze Theorem
- Limits of functions that agree except at a point
Before you start
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