Reference 5: Facts About Limits from Section 3-1
Suppose that
numbers. Then
and
exist, where
and
are real
Theorem 2.1:
Theorem 2.2:
Theorem 2.3:
Theorem 2.4:
Theorem 2.5: If
is a constant, then
.
Theorem 2.6:
Theorem 2.7: If
, then
If is a positive odd integer,
,
Theorem 2.8: or if is a positive even integer and
then
.
If is a polynomial, then
.
Theorem 3: If is a rational function and is in the domain of , then
.
If
and
, then then
is said to have the indeterminate form . In this case,
Definition:
Theorem 2.7 cannot be used to determine the limit.
If
and
, then
does not exist according to the Definition of the Limit
found in Section 3-1.(Remark: In Section 3-2, the Definition of
Limit gets expanded, with the result that in some cases, we will say
Theorem 4:
that the limit is infinity or negative infinity and will write the
symbol
or
. But this will not
always be the case. That is, sometimes the limit will not exist even
with the expanded Definition of Limit.)
If and are functions whose y-values differ at only one x-value,
One Rule
(that is, the y-values are the same for
but differ at
),
(not in book)
then
.
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