Maths (university) · Part 02
Limits and continuity
What a function does near a point, and when its graph has no breaks.
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- LimitThe limit of f(x) as x approaches a is the value f(x) gets closer and closer to. The function does not need to be defined at a itself.
- One-sided limitA limit where x approaches a from only one side: from the left (x below a) or from the right (x above a). The ordinary limit exists only if both agree.
- Limit lawsLimits of sums, differences, products and quotients behave as you would hope: take each limit, then combine. Quotients need a bottom limit that is not zero.
- Squeeze theoremIf a function is trapped between two functions that have the same limit, it must have that limit too.
- Limit at infinityThe value a function approaches as x grows without bound. For fractions of polynomials, divide top and bottom by the highest power of x.
- ContinuityA function is continuous at a if f(a) exists, the limit at a exists, and the two are equal. You can draw its graph without lifting your pen.
- Types of discontinuityA removable discontinuity is a single hole. A jump has different left and right limits. An infinite discontinuity has the function growing without bound.
- Intermediate value theoremIf f is continuous on [a, b] and takes values with opposite signs at the ends, then it equals zero somewhere between them.
- Epsilon-delta definitionThe limit of f(x) at a is L if, for every tolerance epsilon > 0, there is a distance delta > 0 so that x within delta of a gives f(x) within epsilon of L.