Limits and the epsilon-delta Definition (Algebraic Translation)

Vocabulary flashcards covering key terms and concepts from the lecture on limits, epsilon-delta definitions, and translating a graph intuition into algebra.

Limit

The value that f(x) approaches as x approaches c; written as lim_{x→c} f(x) = l.

Open interval about c

An interval around c on which f is defined, used in the formal limit definition.

Epsilon (ε)

A positive tolerance representing how close f(x) should be to l; used in the limit definition.

Delta (δ)

A positive bound on how close x must be to c to ensure |f(x) - l| < ε.

Absolute value

A function giving distance; |a| measures how far a is from 0; used for |x - c| and |f(x) - l|.

Distance

The numerical gap between two values, given by the absolute value of their difference.

Arbitrarily close

For every ε > 0, f(x) can be made within ε of l by choosing x sufficiently close to c.

Within two units of l

A specific tolerance: |f(x) - l| < 2, used in an example to illustrate the idea.

Example: f(x) = 2x - 1, c = 4, l = 7

Here |f(x) - l| = |2x - 8| = 2|x - 4|; to have |f(x) - l| < ε, require |x - 4| < ε/2.

Delta for this example

From |2x - 8| < ε, deduce |x - 4| < ε/2, so δ = ε/2 is a valid choice to guarantee the limit condition.

Limit definition line (formal)

If f is defined near c, then lim_{x→c} f(x) = l means: For every ε > 0, there exists δ > 0 such that 0 < |x - c| < δ implies |f(x) - l| < ε.

Algebraic translation of a limit

Replacing the distance condition with algebraic inequalities to express the relation in terms of x, c, l, and ε; turning the graph idea into algebra.

Graphical vs algebraic approach

The notes compare intuitive graph-based limits with the precise epsilon-delta algebraic form, aiming to turn intuition into exact algebra.