0️Limits and continuity

Overview

1. Definition of a limit

2. Evaluating limits using algebraic techniques and graphs

3. The squeeze theorem

4. Limits at infinity

Definition of a limit

// The limit of f(x) as x approaches a is L
lim   f(x) = L
x→a

If, for every positive number ε, there exists a positive number δ such that for all x satisfying 0 < |x - a| < δ, we have |f(x) - L| < ε.

The symbol ε (epsilon) represents the "error tolerance" we're willing to accept, and the symbol δ (delta) represents the "distance" from a that we need to be to ensure that the error is within our tolerance level.

Evaluating limits using algebraic techniques and graphs

• Direct substitution

• Factorization

• Rationalization

• L'Hopital's rule

• Graphical analysis

The squeeze theorem

It also known as the sandwich theorem or the prinching theorem. It is for evaluating limits.

Statement

Suppose that f(x), g(x), and h(x) are functions defined on some interval containing a point a (except possibly at a) and that for all x in that interval,

$$g(x) ≤ f(x) ≤ h(x)$$

If lim g(x) = L = lim h(x), then lim f(x) also exists and is equal to L.

Limits at infinity

Limits at infinity are a type of limit that describes the behavior of a function as the input variable x gets very large or very small.