Limits formalize the notion of “approaching”
Finite limits at particular points
Finite limits “at” infinity
We write
We write
lim
lim
→
→
to informally mean that the
value
can be made to
be arbitrarily close to by
requiring to be sufficiently
close to , but not equal to
and to formally mean that so long as is a positive number,
there exists a positive number so that trapping in
, ∪
,
guarantees that
is trapped in
,
.
Infinite limits at particular points
to informally mean that the
value
can be made to
be arbitrarily close to by
requiring to be sufficiently
positively large
and to formally mean that so long as is a positive number,
there exists a number so that trapping in , ∞
guarantees that
is trapped in
,
.
“Infinite limits” “at” infinity
We write
lim
→
We write
"
"
∞
lim
→
to informally mean that the
value
can be made to
be arbitrarily positively large
by requiring to be
sufficiently close to , but not
equal to
"
"
∞
to informally mean that the
value
can be made to
be arbitrarily positively large
by requiring to be
sufficiently positively large
and to formally mean that so long as is a number, there exists and to formally mean that so long as
a positive number so that trapping in
, ∪ ,
exists a number so that trapping in
is trapped in
, ∞ .
is trapped in
, ∞ .
guarantees that
Convention for naming values ,
, and
is a number, there
, ∞ guarantees that
th
adopted from Stewart Calculus 4 ed.
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