Epsilon-Delta Definition of a Limit

Calculus

Finite Limit

This basically means that you can make the distance between and arbitrarily small by taking the distance from to to be sufficiently small

Left Hand Limit

Right Hand Limit

Boundedness by a Constant

You can solve Epsilon delta problems by bounding where is a constant. Since you are dealing with a limit you can expect to be within a small distance of such as a distance of . For a distance of you get where is an arbitrary distance around .

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Laws of Limits

Given

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Infinite Limit

Positive Infinity

(If a limit tends to infinity, it does not exist; is not a number)

Negative Infinity

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Limits at Infinity

Definition

Let be defined on then means that values of can be made arbitrarily close by taking sufficiently large .

Precise Definitions

1. • Let be defined on then s.t.

2. • Let be defined on then s.t.

3. • Let be defined on then s.t.

4. • Let be defined on then s.t.

Asymptotes

The line is a horizontal asymptote of if either or

Slant Asymptotes

Some curves have asymptotes that are oblique, neither horizontal nor vertical.