Concept preview

What you will be able to do

1. 01Apply the limit laws (sum, difference, constant multiple, product, quotient, power and root) to evaluate a limit from the limits of its parts.
2. 02Determine when direct substitution is valid — and recognise when it produces the indeterminate form 0/0.
3. 03Distinguish 0/0 (indeterminate — simplify) from a non-zero-over-zero form (an infinite limit / vertical asymptote).
4. 04Evaluate a 0/0 limit by factoring and cancelling, by rationalising with a conjugate, or by combining a complex fraction — then verify the result.
5. 05Explain why cancelling a common factor is legitimate inside a limit even though the function is undefined at the point itself.

A limit is a question about the neighbourhood of a point, not the point itself. When we write $\lim_{x\to c} f(x)$ we are asking: as $x$ gets arbitrarily close to $c$ — from both sides, but never actually equal to $c$ — what single number is $f(x)$ closing in on? The value $f(c)$ might not even exist, and that is fine. The limit can exist where the function does not, and the function can exist where the limit does not. Keeping these two questions separate is the single most important habit in this unit.