Math
115
Notes
'i
LIMITS--Notation
*A
limit
is an intended destination
(there
can
only
be
one
& only
one
destination)
-For
a function, it will be
an
intended f(x)
value
(y:f(x),
so it is an
intended
level for
the
gaph)
*x
approaches an number a
is written as
x-+a
(a
is
a real
number on the
x-uis)
-x
can
approach
a from the right
side of
a
(values
greater
than
a)
written
x-+a*or
from the
left side
of
a
(values
less than a)
wriuen
x--+a-
-f(x)
must
approach
the same
value as x approaches
a from
both
sides
of a.
-x
approaches
infinity is
written as x_+@
(take
x values
to the
exfeme
right
on
the
x-axis)
-x
approaches
negative infinity
(x"'+-o) (take
x
values
to
the extreme
left
on the
x-axis)
-(x)
approaches
a limit
((x)
or
y
values approach
a
fixed level
from
both
sides of :ra)
-(x)
approaches infinity
((x)
or
y
values increase
without
bound
upward)
(curve
goes
up)
-f(x)
approaches negative
infinity
(f(x)
ory
values
decrease
without
bound)
(cyrve
goes
down)
Different
forms
for limits
-The
limit
of
(x)
as x approaches
a
equals L is
written
as
lim
f(x)
=
L
-Right
& left sided limits
are
written
similarly using
rh"
fi;i
and
-
signs
on
a
-The
limit
of f(x) as x approaches
a
is
*
or
-
infinity
(does
not
exist
since
a limit
must be
a real #)
is
written
as
lim f(x):
*o
or -o
(vertical
asymptote)
x+a
-The
limit
of
(x)
as x approaches
infinity
is L is
ranitten
as lim
f(x):
L
(horizontal
asymptote)
-For
a awtoaching
-o, use
x+-co
in above
x+@
-For
a limit
to exist, the right
side limit
and the
left
side limit
must
exist
and
must be
the sane.
-A
limit
does
not exist, ifthe
following
properties
occur:
1.
the
right
side limit and
the Ieft
side
limit
are
different (piecewise
defined
functions)
2.
*
q
or
-@
is
the result
(non-zero
number
over
0)
3.
f(x)
continuously
oscillates
(5sinx
as
x approaches
infinity)
