EC-18.2 Limits and Continuity
You can evaluate a limit, say when one fails to exist and why, and use continuity to argue that a solution has to lie somewhere inside an interval.
Before:nothing requiredUnlocks:EC-3. Differential Equations
This is the topic most readers want to skip and the one that decides whether the rest makes sense. The derivative is a limit and the integral is a limit, so a reader who has only been told the rules can compute but cannot say why a rule stops working. It also carries the first result in this module that an engineer uses directly: if a continuous quantity is negative at one end of an interval and positive at the other, there is a value in between where it is zero, which is the whole idea behind finding a root by halving.
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What a limit says, and why the value at the point is not the question
A limit describes what a function approaches near a point, which is a different question from what it does at the point. Keeping those two apart is the single idea this topic rests on, and most early mistakes come from running them together.
NPTEL: Calculus of One Real Variable (IIT Kanpur) · CourseOne-sided limits, and the ones that do not exist
Approaching from the left and from the right can give different answers, and when they do the limit does not exist. That failure is not a defect in the function; it is a description of a step or a jump that matters physically.
NPTEL: Basic Calculus for Engineers, Scientists and Economists (IIT Kanpur) · CourseThe algebra of limits, and the indeterminate forms
Limits add, multiply and divide the way you would hope, until the parts approach zero or grow without bound together. Those combinations carry no answer on their own and have to be resolved by rewriting the expression first.
Limits at infinity, and describing behaviour far from the origin
Asking what happens as the input grows without bound is how the long-run behaviour of a model is described. Comparing which part of an expression dominates there is a habit that returns in stability arguments much later.
MIT 18.01SC: Single Variable Calculus · CourseContinuity at a point and continuity on an interval
A function is continuous at a point when the limit there exists and equals the value there. Continuity on a whole interval is a much stronger statement, and it is the one every theorem in the next two topics quietly requires.
NPTEL: Calculus of One Real Variable (IIT Kanpur) · CourseThe intermediate value theorem, and finding a root by halving the interval
A continuous quantity that is negative at one end and positive at the other has to be zero somewhere between. Halving the interval repeatedly turns that guarantee into a method that converges, which is the first numerical algorithm in this module.
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