Properties of Limits
If b and c are real numbers, n is a positive
integer, and the functions and g have limits as
, then the following properties are true.
|1. Scalar multiple:||[b((x))] = b[(x)]|
|2. Sum or difference:||[(x)g(x)] = (x)g(x)|
|3. Product:||[(x)g(x)] = [(x)][g(x)]|
|4. Quotient:||[(x)/g(x)] = [(x)]/[g(x)], if g(x)0|
|(x)||x approaches c from the right|
|(x)||x approaches c from the left|
Limits at Infinity
The value of (x) approaches L as x increases/decreases without bound.
y = L is
the horizontal asymptote of the graph of .
Some Nonexistent Limits
Some Infinite Limits
|Exercise:||What is ?|
|(E) The limit does not exist. /TD>|
|The answer is A.||You should memorize this limit.|
A function is continuous at c if:
1. (c) is defined
2. (x) exists
3. (x) = (c)
Graphically, the function is continuous at c if a pencil
can be moved along the graph of (x) through (c,
(c)) without lifting it off the graph.
|Exercise:||If for x 0|
|and if is continuous at x = 0, then k =|
|The answer is E.||(x) = 3/2|
Intermediate Value Theorem
If is continuous on [a, b] and k is
any number between (a) and (b), then
there is at least one number c between a and b
such that (c) = k.