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
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.