EC-18.2 Limits and Continuity
The standard treatment of limits, continuity and the intermediate value theorem in an engineering calculus course, September 2026
What this is and why it exists
This is the topic most readers want to skip, and the one that decides whether the rest makes sense.
The derivative is a limit. The integral is a limit. An infinite series is a limit. A reader who has only been told the rules can compute correctly. What they cannot do is say why a rule stops working, and so they cannot tell when it has.
It also carries the first result in this module an engineer uses directly. Take a continuous quantity that is negative at one end of an interval and positive at the other. There is a position between them where it is zero. That is the whole idea behind finding a root by halving.
The vocabulary
- Limit — the value a function approaches as its input approaches a position.
- One-sided limit — the same question asked from the left only, or from the right only.
- Indeterminate form — a combination such as zero over zero, where the parts alone decide nothing.
- Limit at infinity — what the function approaches as the input grows without bound.
- Asymptote — a line the graph approaches without meeting.
- Continuous at a position — the limit there exists and equals the value there.
- Continuous on an interval — continuous at every position inside it.
- Intermediate value theorem — a continuous function on an interval takes every value between its two end values.
- Bisection — halving an interval repeatedly to trap a root.
The mental model
Hold one distinction and most of this topic follows. A limit is about what happens near a position, and says nothing about what happens at it.
A function can be undefined at a position and still have a perfectly good limit there. It can be defined at a position, have a limit there, and have the two disagree. Running the two questions together is where nearly every early mistake comes from.
The reason this matters is the derivative. The derivative at a position is the limit of a ratio whose lower part goes to zero. At the position itself that ratio is undefined. If the limit could not exist without the value, there would be no derivative at all.
One-sided limits come next. Approaching from the left and from the right can give different answers, and when they do, the limit does not exist. This is not a defect in the function. It is a description of a step, and steps are physical. A switch closing is a step, and the current in an ideal switch has no limit at the instant it closes.
The algebra of limits behaves the way you would hope. Limits add, multiply and divide, provided the pieces exist and you are not dividing by zero. The exceptions are the indeterminate forms, where both parts head to zero or both grow without bound. An indeterminate form is not an answer and not a failure. It is an instruction to rewrite the expression and ask again.
Limits at infinity describe long-run behaviour. The practical skill is comparing which part of an expression dominates far from the origin. In a ratio of polynomials, the highest power on each side settles the answer and everything else fades. That habit returns in stability arguments much later.
Continuity is then short to say. A function is continuous at a position when the limit there exists and equals the value there. Continuity on a whole interval is a much stronger statement, and it is what every theorem in the next two topics quietly requires.
The intermediate value theorem is the payoff. A continuous quantity that is negative at one end of an interval and positive at the other must be zero somewhere between. Nothing here says where. It says only that a position exists.
That existence is enough to build a method. Evaluate at the midpoint. If the sign there matches the left end, the root is in the right half; otherwise it is in the left. Repeat. Each step halves the interval, so the position is trapped ever more tightly, and the procedure never fails when the theorem's conditions hold. It is slow and it always works, which is a combination worth respecting.
What you should now be able to explain or do
- Evaluate a limit, and say why the value at the position is a separate question.
- Recognise when one-sided limits disagree, and say what physical situation that describes.
- Identify an indeterminate form and rewrite the expression instead of stopping.
- Compare growth rates to find a limit at infinity.
- State what continuity at a position requires, and what continuity on an interval adds.
- Use the intermediate value theorem to trap a root, and carry out a bisection.
Check yourself
A function is undefined at a position. Can it still have a limit there?
Yes, and this is the ordinary case rather than the odd one. A limit describes behaviour near the position. The derivative exists only because a ratio that is undefined at a position can still have a limit there.
The left and right limits at a position give different values. What follows?
The limit does not exist there. That is a description of a step in the function, and steps are physical rather than pathological. A switch closing produces one.
An expression evaluates to zero over zero. What have you learned?
That the form alone decides nothing. Both parts approach zero, and the answer depends on how fast each does. Rewrite the expression, by factoring or by another route, and ask again.
Why does bisection always work when its conditions hold?
Because the intermediate value theorem guarantees a root inside an interval whose ends have opposite signs. Halving keeps a root trapped in the surviving half, so the interval shrinks towards it without fail.
What does continuity on an interval give you that continuity at one position does not?
The theorems. The intermediate value theorem, the mean value theorem and the results behind integration all need continuity across a whole interval, not at a single position.
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