Limits

Properties of Limits

If b and c are real numbers, n is a positive integer, and the functions ƒ and g have limits as , then the following properties are true.

1. Scalar multiple:	[b(ƒ(x))] = b[ƒ(x)]
2. Sum or difference:	[ƒ(x)g(x)] = ƒ(x)g(x)
3. Product:	[ƒ(x)g(x)] = [ƒ(x)][g(x)]
4. Quotient:	[ƒ(x)/g(x)] = [ƒ(x)]/[g(x)], if g(x)0

One-Sided Limits

ƒ(x)	x approaches c from the right
ƒ(x)	x approaches c from the left

Limits at Infinity

ƒ(x) = L

or

ƒ(x) = L

The value of ƒ(x) approaches L as x increases/decreases without bound.

y = L is the horizontal asymptote of the graph of ƒ.

Some Nonexistent Limits

sin

Some Infinite Limits

ln x

Exercise:	What is ?
	(A) 1
	(B) 0
	(C)
	(D)
	(E) The limit does not exist. /TD>
The answer is A.	You should memorize this limit.

Continuity

Definition

A function ƒ is continuous at c if:

1. ƒ(c) is defined

2. ƒ(x) exists

3. ƒ(x) = ƒ(c)

Graphically, the function is continuous at c if a pencil can be moved along the graph of ƒ(x) through (c, ƒ(c)) without lifting it off the graph.

Exercise:	If for x 0
	and if ƒ is continuous at x = 0, then k =
	(A) -3/2
	(B) -1
	(C) 0
	(D) 1
	(E) 3/2
The answer is E.	ƒ(x) = 3/2

Intermediate Value Theorem

If ƒ is continuous on [a, b] and k is any number between ƒ(a) and ƒ(b), then there is at least one number c between a and b such that ƒ(c) = k.