;
-The
following curve
has
an
infinite
discontinuity
at
ra.
This means
f(x)+co
tts
x-)4.
There is
no limit
in
this
case.
The
vertical line:<=a
is
called
a
y!:rtical
asvmutoE.
@
For a function
to
be
continuous at
)FL
the
following
limit must
hold:
lim
{x)
=
f(a). That
is, the
limit must
exist
as
x-+4,
the
y
value,
(a}
must be
x-+a
defined,
and,
most
importantly,
the limit
must
=
{a).
These
are
sometimes
referred to the
tkee
i's
of contimrity
in
a
formal calculus
course.
-Look
back
to
the
3
graphs
with discontinuities
already
covered.
What
part(s)
of
this limit
staternent
fail?
-If
a curve
is
continuous,
then
you
will
be able
to
trace
the
entire
Saph
without
lifting
your
pen
or
pencil
offthe
paper.
Pronerties
of
limits
1) lim K =
K
(limit
of a coostant
is itsel{
no matter
what
x approaches,
xrla
2) tim
t(x)*g(x)]
=
lim
(x)
*
lim
g(x) (limit
of
a
srm or
difference
is the
sum or
x-)a
x-)a
x-+a
difference
ofthe
limits)
3) lim
t{x)S(x)l
=
[ti*
(")]
[1i*
g(x)]
([mit
of
a
product
is
the
product
ofthe
limits)
x->a
x*+a
x-)a
4)
lim
[(*ys6)]=
[ri*
f(-)]l[ri*
gt.l]
(tmit
of a
quotient is
the
quotient
of
the limits,
x-)a
5)
Iim
x
x-+a
x*+a
x*)a
provided
that
the bottom
limit
is
not
zero)
'
=
au,
for
n
>
o-
(power
property)
*All
ofthe
above assume
that the
limits
lim f(x) and
lim
g(x)
exist.
x-+a
x-+a
-These
are
also referred
to
the
*limit
theorems"
in
a
formal calculus
course.
