3.1 - MATH
1081
 HOMEWORK
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Unformatted text preview: MATH
1081
 HOMEWORK
HEADER
 
 Your
name
 Your
TA’s
name
 Your
recitation
number
 The
assignment
number
 
 Multiple
pages
MUST
be
stapled.
 MATH
1081
 Monday,
January
24
 
 Chapter
3
–
Section
1
 
 LIMITS
 
 Homework
#2
(due
1/31):
 Section
3.1
#8,
34,
38,
40,
42,
46,

 48,
50
 
 Clicker
Check­in:

Choose
any
letter
to
check
in
now.
 In
order
to
define
the
central
ideas
of
calculus,
we
need
the
 notion
of
something
called
a
limit.

The
basic
idea
of
a
limit
is
to
 answer
the
question
“What
happens
to
the
output
values
of
the
 function
 f ( x ) 
as
the
input
 x 
moves
towards
some
designated
 point?”
 
 As
we
will
see,
the
answer
to
this
question
can
be

 • that
the
output
values
get
closer
and
closer
to
a
specific
 number,

 • that
the
output
values
are
unbounded
go
off
to
infinity,
or

 • that
there
is
just
no
single
answer
to
this
question. 

 Shown
here
is
the
graph
of
some
function
 f ( x ) .
 
 
 According
to
the
graph,
what
is
 f (1) ?

 f ( 0 ) ?

 f ( −2 ) ?
 
 Now,
consider
this
same
function.

 
 
 
 What
value
is
the
function
tending
towards
as
 x 
gets
closer
and
 closer
to
1?
 
 What
about
as
 x 
gets
closer
and
closer
to
0?

or
‐2?
 
 When
we
find
the
value
of
a
limit,
we
are
asking
the
question
 “As
 x 
gets
closer
and
closer
to
some
specific
value
 a ,
where
are
 the
values
of
 f ( x ) 
going?”
 
 It
turns
out
that
the
answer
to
the
question
does
not
really
 depend
on
what
is
actually
happening
when
 x = a ,
because
in
a
 limit
we
are
only
asking
what
happens
as
 x 
gets
close
to
 a .
 
 
 The
notation
we
use
is
 
 
 
 
 
 
 
 lim f ( x ) = L 
 
 which
says
that
the
values
of
 f ( x ) 
tend
towards
the
value
 L 
as
 x 
gets
closer
and
closer
to
 a .
 
 
 So,
for
the
function
 f ( x ) 
we
saw
in
the
graph,

 
 1 
 
 
 
 lim f ( x ) = 2 
 and
 lim f ( x ) = − .
 x →1 x→ 0 2 
 Note
that
 f (1) = 2 ,
but
 f ( 0 ) 
does
not
exist.
 x→ a For
this
same
function,
what
can
we
say
about
 lim f ( x )?
 x → −2 
 
 
 There
are
two
issues:
 1. It
depends
if
we
let
 x 
get
closer
and
closer
to
‐2
from
 the
left
side
or
from
the
right
side.
 2. Even
if
we
only
look
at
one
side,
to
what
numerical
 value
are
the
values
of
 f ( x ) 
tending?
 To
address
the
first
issue,
we
can
use
a
one­sided
limit,
which
 (as
the
name
implies)
focuses
 x 
to
get
closer
to
 a 
from
only
one
 side.
 
 The
notation
that
we
use
is
 
 
 lim− f ( x ) = L 

to
denote
 x 
getting
closer
to
 a 
from
the
left,

 x→ a 
 and
 
 
 
 lim+ f ( x ) = L 

to
denote
 x 
getting
closer
to
 a 
from
the
right,

 x→ a so
 x < a 
 so
 x > a .
 Note
that
if
these
two
one‐sided
limits
don’t
agree
(so
our
 answer
depends
on
which
side
of
 a 
we
look),
there
is
no
 answer
to
the
regular
(two‐sided)
limit
question.
 
 That
is,
when
 
 
 
 
 
 lim− f ( x ) ≠ lim+ f ( x ) 
 
 then
 


 x→ a x→ a 
 
 lim f ( x ) 
does
not
exist. x→ a To
address
the
second
issue,
we
will
abuse
the
notation
and
 write
 lim f ( x ) = ∞ 
(or
 −∞ )
to
indicate
that
the
values
are
 unbounded.


 
 Since
infinity
is
not
a
real
number,
the
value
of
the
limit
in
such
 a
case
actually
does
not
exist.


However,
because
writing
 lim f ( x ) = ∞ 
tells
us
what
is
happening
to
the
function,
we
will
 use
it
when
we
can.
 
 So,
for
the
function
we
were
examining,
we
would
write
 lim− f ( x ) = +∞ 
 and
 lim+ f ( x ) = −∞ 
 


 
 and
because
these
don’t
agree,
 lim f ( x )
does
not
exist.
 x → −2 x→ a x→ a x → −2 x → −2 Example:
 The
graph
of
the
function
 h ( x )
is
shown
here.

 
 
 
 Determine
the
value
of
each
limit,
if
it
exists.
 
 
 
 
 
 
 
 
 
 
 
 
 
 1.

 lim h ( x ) 
 x→ 2 
 
 
 
 
 
 
 
 
 3.
 lim+ h ( x )
 x→ 4 2.
 lim h ( x ) 
 x→ 4 4.
 lim− h ( x )
 x→ 4 This
figure
is
from
page
155
in
your
textbook
which
shows
 every
possible
possibility
when
evaluating
a
limit
with
a
graph.
 
 As
we
have
seen,
the
value
of
the
limit
 lim f ( x )
and
the
value
 x→ a of
the
function
 f ( a ) 
are
not
necessarily
related
in
any
way.
 
 The
limit
can
exist
when
the
value
of
the
function
does
not
 exist,


 OR
 the
value
of
the
function
can
exist
when
the
limit
does
not,

 OR
 they
can
both
exist
but
simply
not
be
equal.
 
 The
case
where
both
exist
and
are
equal
to
one
another
is
of
 particular
interest
because
it
implies
that
at
that
point,
there
is
 no
hole,
no
gap,
no
asymptote.

We
will
revisit
this
in
Section
 3.2.
 In
addition
to
examining
the
behavior
of
a
function
near
 certain
values
of
 x ,
it
is
sometimes
useful
to
examine
the
 behavior
of
the
function
as
 x 
gets
larger
and
larger
(or
smaller
 and
smaller).
 
 For
example,
suppose
we
are
analyzing
the
population
of
a
 species
over
time.

We
may
want
to
consider
what
will
happen
 far
into
the
future,
according
to
the
model
for
recent
 population
changes.
 
 In
this
case,
we
are
looking
at
what
happens
to
the
function
out
 towards
the
edges
of
the
graph.

Is
there
some
clear
trend?
 
 So,
we
are
asking
the
question
“As
 x 
gets
larger
and
larger,
 what
do
the
values
 f ( x ) 
tend
towards?”
 
 The
notation
we
use
is
 
 
 
 
 
 lim f ( x ) = L 
 and
 lim f ( x ) = M .
 x→ ∞ x→ − ∞ 
 For
many
of
the
functions
we
work
with,
these
two
limits,
if
 they
exist,
will
be
equal.


 However,
there
is
no
reason
that
they
have
to.
 
 Note
that
when
the
values
of
a
function
to
tend
towards
a
 specific
value
as
 x 
tends
towards
the
infinities,
it
is
typical
to
 indicate
this
on
a
graph
using
a
horizontal
asymptote.
 
 For
example,

 indicates
 lim f ( x ) = lim f ( x ) = 3 .
 x→ ∞ x → −∞ 
 Clicker
Check­out:

Choose
any
letter
to
check
out
now.
 
 
 Next
time
–
Section
3.1
continued.
 
 
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