Calculus 2.2 Limits of a functions
Definition of limit: Suppose f(x) is defined when x is near the number a. (This means that f is defined on some open
interval that contains a, except possibly at a itself.) Then we write:
lim
→
And say, “The limit of f(x), as x approaches a, equals L,”
If we can make the values of f(x) arbitrarily close to L (as close as to L as we like) by taking x to be sufficiently close
to a (on either side of a) but not equal to a.
We can find a limit by finding values of f near a and “guess” the limit. (There are algebraic methods as well that we
will learn in the next few days!)
Look at a graph of the function and “guess” the limit by examining points near the value of a.
2. lim →
1. lim →
3. lim
→
√
4. lim
→
One-sided limits: Some functions only have limits from one side, or have different limits from each side.
Lefthand limit: lim →
Righthand limit: lim
→
5. The Heavyside function:
0
1
0
0
Right side:
Left side:
If the lefthand limit as x approaches a does not equal the righthand limit as x approaches a, we say that the limit as x
approaches a does not exist. If the lefthand limit as x approaches a equals the righthand limit as x approaches a, we
say that the limit as x approaches a exists.
If lim
lim
→
→
, then lim
→
Infinite limits: Let f be defined on both sides of a, except possibly at a itself, Then lim →
∞ means that the
values of f(x) can be made arbitrarily large (as large as we please), or arbitrarily large negative by taking x sufficiently
close to a, but not equal to a.
Examine the following limits. Which ones go to positive or negative infinity?
1
7. lim →
6. lim
→
8.
lim
→
Definition: The line
statements is true:
lim
→
lim
→
is called a vertical asymptote of the curve
∞
lim
→
∞
9. Find the vertical asymptotes of
using the concept of an infinite limit.
lim
→
tan
∞
∞
if at least one of the following
lim
→
lim
→
∞
∞
10. Find lim → ln (Why don’t we find the limit
from the left side?)