Limits and Continuity

Stuff to know from Unit 1

What does continuous mean?

going on without a stop or break

Limit Existence Theorem

The limit as x approaches c on f(x) will exist if and only if the limit as x approaches c from the left is equal to the limit as x approaches c from the right.

Three Part Definition of Continuity

3 Possible limits of an exponential function?

infinity, negative infinity, Horizontal Asymptote

lim sincx/cx x->0

1

• lim cos(cx) - 1 / cx x-> 0

• lim 1 - coscx/cx

0

Intermediate Value Theorem (IVT)

if f(x) is continuous on [a, b] and f(a) < y < f(b) or f(a) > y > f(b), then there exists at least one value, x = c on (a, b) such that f(c) = y

IVT (laymen's term)

If the function is continuous between the x-values of a and b, then there is a x-value (c) that is between a and b (not included) with a y-value

Two conditions to verify IVT

1. 1. f(x) must be continuous at [a, b]

2. 2. f(c) must be between f(a) and f(b)

Infinite Limits

NOTE: Look for Vertical Asymptotes

Limits at Infinity

Answers will be either:

• End Behavior (+ - Infinity)

• Horizontal Asymptotes (Number) (NOTE: rational functions have more than 1 HA)

• Slant Asymptote

Justification of the existence of a Vertical Asymptote Using Limits

How to Algebraically Evaluate a limit at infinity for rational functions

1. Divide numerator and denominator by degree of denominator

2. Evaluate (Cancel out)