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What you will learn in Limits & Continuity
1. What a Limit Is
The value a function heads for, which is a different question from the value it takes — and they can disagree.
2. Limits from Tables and Graphs
Estimating a limit by approaching from each side, and seeing why the hole never fills however far you zoom.
3. Algebraic Evaluation of Limits
Three techniques for the case where substitution gives 0/0, and the one sentence that makes all three legitimate.
4. One-Sided Limits and Existence
Two approaches, run separately, and the two-part condition for the two-sided limit to exist at all.
5. Infinite Limits and Vertical Asymptotes
When the values run away, and why a zero denominator does not always mean a wall.
6. Limits at Infinity and End Behaviour
What happens far from the origin, decided by a degree comparison — and the range where the loser is winning.
7. Standard Limits
Six results worth knowing, each with its origin — and the one that explains why calculus uses radians.
8. The Squeeze Theorem
Trapping a function between two others that meet — and the condition learners routinely over-state.
9. Continuity at a Point and on an Interval
Three conditions checked in order, and the theorem that finally licenses substitution.
10. Types of Discontinuity
Removable, jump, infinite and oscillatory — four different failures, and only one of them can be mended.
11. The Intermediate Value Theorem
Continuity guarantees a root exists. Finding it is a separate job, and bisection does it by halving.
12. Continuity and Differentiability
The corner that separates them, and the picture module 18 begins from.