Limits
A function can be undefined at a point and still head straight toward one height there. How do you name that height, compute it, and use it to say what a function does near a gap, at infinity, or across an interval?
Where you are 11 inside
What this area is about
The limit \lim_{x \to a} f(x) = L says the outputs come as close to L as you like for inputs close enough to a, with x \ne a. It ignores the value at a itself, which is exactly what derivatives and integrals need: both are limits of quantities that cannot be computed at the point.
How these fit together
Meaning. What a limit is reads limits off graphs and tables. One sided limits split the approach into left and right. The epsilon delta definition makes “as close as you like” exact.
Computing. The limit laws combine known limits. Factoring and rationalizing rewrite a quotient that gives \displaystyle \frac{0}{0}. The squeeze theorem traps a function between two others with the same limit, and proves \displaystyle \lim_{x \to 0} \frac{\sin x}{x} = 1 for x in radians.
Infinity. Infinite limits locate vertical asymptotes, limits at infinity horizontal ones, and growth rates rank logarithms, powers and exponentials.
Continuity. Continuity asks that the limit equal the value. The intermediate value theorem says that if f is continuous on [a, b], it takes every value between f(a) and f(b).
A path through it
Graphs and tables, then one sided limits, then the limit laws. Factoring before the squeeze theorem, which refers back to it. Infinite limits, then limits at infinity. Continuity needs the laws, one sided limits and infinite limits; the intermediate value theorem needs continuity. Growth rates use limits at infinity and continuity. The epsilon delta definition needs only graphs, but it looks back at most of the area, so it fits best last.
Next come the derivative and the definite integral.