limits - 2A: Lecture 3 
 Section 2.2 The limit of a...

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Unformatted text preview: 2A: Lecture 3 
 Section 2.2 The limit of a function 
 
 Group activity: Suppose
f
is
a
function
defined
 around
a
point
a.

 
 We
say
that


lim f(x)=L if… x a 
 2
 Informal
definition:

 
 We
say
that
the
limit
of
f(x)
as
x
approaches
a

 is
equal
L
and
we
write
 
 

























 
 
 if
we
can
make
f(x)
as
close
to
L
as
we
want

 by
taking
x
sufficiently
close
to
a,
but
not
equal
to
a.
 
 
 3
 
 
 
 lim f(x) = 3 x -2 TRUE FALSE 
 4
 
 lim f(x) = 3 x -2 TRUE FALSE If x approaches -2 (from either direction), f(x) approaches 3. We can make f(x) as close to 3 as we want, by taking x sufficiently close to -2 (but not equal to -2). So lim f(x)=3. x 
 -2 5
 Note that f(-2)=2, not 3, but this does not matter at all. lim f(x)= - 4 x 3 TRUE FALSE 
 6
 
 
 lim f(x)= - 4 x TRUE FALSE 3 If x approaches 3 from the left, f(x) approaches 0. If x approaches 3 from The right, f(x) approaches -4. So lim f(x). x 
 3 7
 does not exist. 




E 
 
 
 
 
 
 Choose
values
of
x
that
get
closer
and
 closer
to
x=2
(but
are
not
=2)
and
plug
 these
values
into
the
function.
 
 x >2 2 .5 2 .1 2 .0 1 2.001 2.0001 2.00001 
 f(x ) 3 .4 3.857142857 3.985074627 3.998500750 3.999850007 3.999985000 x <2 1 .5 1 .9 1 .9 9 1.999 1.9999 1.99999 f(x ) 5 .0 4.157894737 4.015075377 4.001500750 4.000150008 4.000015000 Take
a
guess!
 
 The
limit
of
f(x)
as
x
approaches
2
is
…
 

 
 
 
 8
 
 





 x >2 2 .5 2 .1 2 .0 1 2.001 2.0001 2.00001 
 f(x ) 3 .4 3.857142857 3.985074627 3.998500750 3.999850007 3.999985000 x <2 1 .5 1 .9 1 .9 9 1.999 1.9999 1.99999 f(x ) 5 .0 4.157894737 4.015075377 4.001500750 4.000150008 4.000015000 
 The
values
of
f
approach
4.

 
 We
guess
that



 
 
 














OK!
 

































































 
 
 9
 

 








 
 
 Choose
values
of
x
that
get
closer
and

 closer
to
x=2
(but
are
not
=2)
and
plug
 these
values
into
f.
 
 
same
table
of
values
as
before!

 
 GUESS:



 










 Limits
are
concerned
with
what
goes
on
 around
the
point
(not
at
the
point
itself!)
 
 10
 
 
f
is
not
defined
at
o !
 Choose
values
of
x
that
get
closer
and
 closer
to
x=0
(but
are
not
=0)
and
plug
 them
into
the
function.
 >0 1 0 .1 0 .0 1 0.001 0.45969769 0.04995835 0.00499996 0.00049999 <0 -1 -0.45969769 - 0 .1 -0.04995835 - 0 .0 1 -0.00499996 -0.001 -0.00049999 
 The
values
of
f
approach
0.

 
 We
guess
that




 
 OK
 
 
 11
 Can
we
always
guess
the
limit
 this
way?

 
 12
 
 
 



 
 

f
is
not
defined
at
o !
 Choose
values
of
x
that
get
closer
and
 closer
to
x=0
(but
are
not
=0)
and
plug
 these
values
into
f. 
 t 1 0 .1 0 .0 1 0.001 f(t) -1 1 1 1 t -1 - 0 .1 - 0 .0 1 -0.001 f(t) -1 1 1 1 
 
 
 Guess:
the
limit
is
1.

 
 
 
 
 13
 
 This
guess
is
quite
WRONG!!!
 
 
 
 f
oscillates
rapidly
around
0,
and
never
 settles
down
to
a
fixed
value.

 
 The
limit
of
f
as
x
approaches
0
does
not
 exists!
 
 
 
 
 
 
 
 14
 
 
 
 (Heaviside
or
step
function) 
 
 The
function
doesn’t
settle
down
to
a
 single
number
as
we
move
in
towards
0.
 
 If
we
approach
0
from
the
right,
the
 values
of
f
move
toward
1.
 If
we
approach
0
from
the
left,
the
values
 of
f
move
toward
0.
 
 The
limit
of
f
as
x
approaches
0
DNE. 
 15
 Limit a
 We
say
that
the
limit
of
f
as
x
approaches
a
is
equal
to
 L,
and
we
write
 





























 
 if
we
can
make
f(x)
as
close
to
L
as
we
want

 by
taking
x
sufficiently
close
to
a
but
different
from
a. 
 Right-handed limit a We
say
that
the
limit
of
f
as
x
approaches
a
from
the
 right
is
equal
to
L
and
we
write
 






























 
 if
we
can
make
f(x)
as
close
to
L
as
we
want

 by
taking
x
sufficiently
close
to
a
but
bigger
than
a. 
 Left-handed limit a We
say
that
the
limit
of
f
as
x
approaches
a
from
the
 left
say
that
the
limd
wfe
ws
xe
 pproaches
a
from
the
 We 
is
equal
to
L
an it
o 
f
a rit 
a 
 eft
is
equal
to
L
and
we
write
 l 
 






























 
 if
we
can
make
f(x)
as
close
to
L
as
we
want

 






























 
 by
te
cian
makff
ifcients
cloose
toLa
but
e
wanlte
r
than
a. if
w ak ng
x
sue (x)
a ly
cl se
to
 
 
as
w smal 
 by
taking
x
sufficiently
close
to
a
but
smaller
th6an
a. 
 1
 
 
 
f
is
not
defined
at
‐4.

 The
limit
exists:

 
 
 f
is
defined
at
+1.

 The
limit
does
not
exist:
 
 
 
 







 







 
 f
is
defined
at
+6.


 The
limit
exists:


 
 
 17
 f(2) lim f(x) x2 + f(3) lim f(x) x3 + f(1) lim f(x) x1 + lim f(x) x2 x2 - lim f(x) x3 x3 - lim f(x) x1 x1 - lim f(x) lim f(x) lim f(x) 
 18
 f(2) 4 lim f(x) 4 x2 + f(3) ~2.5 lim f(x) 1 x3 + f(1) ~1.5 lim f(x) 2 x1 + lim f(x) 4 x2 x2 - lim f(x) ~2.5 x3 x3 - lim f(x) 2 x1 x1 - lim f(x) 4 lim f(x) DNE lim f(x) 2 
 20
 
 Choose
values
of
x
that
get
closer
 and
closer
to
x=0
(but
are
not
=0)
 and
plug
these
values
into
f.
 x - 0 .1 - 0 .0 1 -0.001 -0.0001 
 1/x -1 0 -1 0 0 -1 0 0 0 -10000 x 0 .1 0 .0 1 0.001 0.0001 1/x 10 100 1000 10000 
 As
we
make
x
smaller
and
smaller,
 f(x)=
1/x
gets
larger
and
larger

 but
it
retains
the
same
sign
as
x

 
 Guess:
The
limit
does
not
exist.
 
 
 23
 
 We
say
that
x=0
is
a
vertical
 asymptote
for
the
graph.
 
 
 
 
 
 
 24
 
 
 
 
 Again
the
limit
is
infinite.
 
 We
say
that
x=0
is
a
vertical
asymptote

 for
the
graph.
 
 25
 
 
 
 
 
 Infinite Limit We say that the limit of f as x approaches a is + infinity, and we write if we can make f(x) arbitrarily large, by taking x sufficiently close to a, but different from a. We say that the limit of f as x approaches a is -infinity, and we write if we can make f(x) arbitrarily large, by taking x sufficiently close to a, but different from a. 
 
 The
definition
of
right
and
left
infinite
limits
is
similar.
 
 26
 Vertical Asymptote The function f(x) has a vertical asymptote at in any of the following cases: 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 27
 x=‐2
is
a
vertical

asymptote.
 
 
 
 
 x=+3
is
a
vertical

asymptote.
 
 
 
 

 x=0
is
a
vertical

asymptote 
 28
 

 The
tangent
function
has
many
vertical
 asymptotes...
 
 
 
 
 
 29
 3
WAYS
IN
WHICH
THE
LIMIT
MAY
FAIL
TO
EXIST 
 
 The limit from the left is different from the limit from the right.
 
 a
 
 
 
 


The
limit
(from
the
 right
or
from
the
left)
does
not
exist
 because
the
function
oscillates.
 
 

The
limit
is
infinite.
 
 30
 ...
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This note was uploaded on 12/07/2010 for the course MATH MATH 2A taught by Professor Alessandrapantano during the Fall '10 term at UC Irvine.

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