The Precise Definition of a Limit

From intuition to precision

Verifying simple limits

Delta as a function of epsilon

The definition's descendants

In the epsilon-delta definition of limit, which quantity is chosen first and which may depend on it?

Why does the definition impose 0 less than |x - a| rather than |x - a| less than delta?

To prove lim x to 3 of (2x + 1) equals 7 by the definition, we need |f(x) - 7| less than epsilon. What choice of delta works?

Verifying lim x to 2 of x^2 = 4 requires bounding |x + 2|. Why precommit to delta at most 1 before choosing the final delta?

Why does the epsilon-delta definition certify that 1/x has no limit as x approaches 0?

The definition says for every epsilon there exists a delta. What happens to the definition if the quantifiers were reversed, for every delta there exists an epsilon?

The precise definition of limit is known by the names of the two Greek letters it uses. What is it called, in lowercase with a hyphen? Answer like alpha-beta.

In the definition, the condition |f(x) - L| less than epsilon says the outputs lie within epsilon of L. What single Greek letter names that tolerance? Answer with the letter name spelled out.

Verifying the limit of x^2 at 2 with delta at most 1 bounds |x + 2| by a specific integer. What integer? Answer with a number.

Continuity's definition relaxes one feature of the limit definition by allowing x equal to a. How many strict inequalities of the form 0 less than |x - a| appear in the continuity version? Answer with a number.