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L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0008.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hopi= tal's Rule: Given lim f(x) =3D lim g(x) =3D 0, then
then l= im
f(x)
g(x)
=3D
lim
f '(x)=
g'(x)<= /span>
if the limit exists, or its ±∞ .
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0008_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhgwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACB AAEAgAAAAAAAAAIOjI+pu+APo5y02seyzgUAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0009.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hopi= tal's Rule: Given lim f(x) =3D lim g(x) =3D 0, then
then l= im
f(x)
g(x)
=3D
lim
f '(x)=
g'(x)<= /span>
if the limit exists, or its ±∞ .
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
We cal= l this type of limits as the
"= ; 0 "
0
indet= erminate
forms.=
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0009_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhgwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACB AAEAgAAAAAAAAAIOjI+pu+APo5y02seyzgUAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0009_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0010.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hopi= tal's Rule: Given lim f(x) =3D lim g(x) =3D 0, then
then l= im
f(x)
g(x)
=3D
lim
f '(x)=
g'(x)<= /span>
if the limit exists, or its ±∞ .
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
We cal= l this type of limits as the
"= ; 0 "
0
indet= erminate
forms.=
L'Hop= ital's rule passes the calculation of the 
"= 0 "
0
form = to the calculation of derivatives.
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0010_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhgwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACB AAEAgAAAAAAAAAIOjI+pu+APo5y02seyzgUAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0010_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0010_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0011.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hopi= tal's Rule: Given lim f(x) =3D lim g(x) =3D 0, then
then l= im
f(x)
g(x)
=3D
lim
f '(x)=
g'(x)<= /span>
if the limit exists, or its ±∞ .
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
We cal= l this type of limits as the
"= ; 0 "
0
indet= erminate
forms.=
L'Hop= ital's rule passes the calculation of the 
"= 0 "
0
form = to the calculation of derivatives.
Example: A. Find lim
sin(x)
x
xà0
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0011_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhgwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACB AAEAgAAAAAAAAAIOjI+pu+APo5y02seyzgUAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0011_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0011_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0011_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAm AAEAgAAAAAAAAAIFjI+py1YAO1== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0012.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hopi= tal's Rule: Given lim f(x) =3D lim g(x) =3D 0, then
then l= im
f(x)
g(x)
=3D
lim
f '(x)=
g'(x)<= /span>
if the limit exists, or its ±∞ .
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
We cal= l this type of limits as the
"= ; 0 "
0
indet= erminate
forms.=
L'Hop= ital's rule passes the calculation of the 
"= 0 "
0
form = to the calculation of derivatives.
Example: A. Find lim
sin(x)
x
xà0
Since lim sin(x) =3D 0 and lim x =3D 0,
xà0
xà0
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0012_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhgwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACB AAEAgAAAAAAAAAIOjI+pu+APo5y02seyzgUAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0012_image012.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0012_image013.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0012_image014.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAm AAEAgAAAAAAAAAIFjI+py1YAO1== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0074.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hopi= tal's Rule: Given lim f(x) =3D lim g(x) =3D 0, then
then l= im
f(x)
g(x)
=3D
lim
f '(x)=
g'(x)<= /span>
if the limit exists, or its ±∞ .
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
We cal= l this type of limits as the
"= ; 0 "
0
indet= erminate
forms.=
L'Hop= ital's rule passes the calculation of the 
"= 0 "
0
form = to the calculation of derivatives.
Example: A. Find lim
sin(x)
x
xà0
Since lim sin(x) =3D 0 and lim x = =3D 0, use the L'Hopital's
rule:
xà0
xà0
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0074_image015.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhgwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACB AAEAgAAAAAAAAAIOjI+pu+APo5y02seyzgUAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0074_image016.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0074_image017.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0074_image018.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAm AAEAgAAAAAAAAAIFjI+py1YAO1== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0013.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hopi= tal's Rule: Given lim f(x) =3D lim g(x) =3D 0, then
then l= im
f(x)
g(x)
=3D
lim
f '(x)=
g'(x)<= /span>
if the limit exists, or its ±∞ .
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
We cal= l this type of limits as the
"= ; 0 "
0
indet= erminate
forms.=
L'Hop= ital's rule passes the calculation of the 
"= 0 "
0
form = to the calculation of derivatives.
Example: A. Find lim
sin(x)
x
xà0
Since lim sin(x) =3D 0 and lim x = =3D 0, use the L'Hopital's
rule:
xà0
xà0
lim
sin(x)
x
=3D
lim
[sin(x)]'
[x]'
xà0
xà0
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0013_image019.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhgwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACB AAEAgAAAAAAAAAIOjI+pu+APo5y02seyzgUAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0013_image020.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0013_image021.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0013_image022.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAm AAEAgAAAAAAAAAIFjI+py1YAO1== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0013_image023.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhswACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACx AAEAgAAAAAAAAAIRjI+pywcPo5y02otl27z7XgAAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0014.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hopi= tal's Rule: Given lim f(x) =3D lim g(x) =3D 0, then
then l= im
f(x)
g(x)
=3D
lim
f '(x)=
g'(x)<= /span>
=
if the limit exists, or its ±∞ .
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
We cal= l this type of limits as the
"= ; 0 "
0
indet= erminate
forms.=
L'Hop= ital's rule passes the calculation of the 
"= 0 "
0
form = to the calculation of derivatives.
Example: A. Find lim
sin(x)
x
xà0
Since lim sin(x) =3D 0 and lim x = =3D 0, use the L'Hopital's
rule:
xà0
xà0
lim
sin(x)
x
=3D
lim
[sin(x)]'
[x]'
xà0
xà0
=
=3D
lim
cos(x)
1
xà0
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
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L'Hopi= tal's Rule: Given lim f(x) =3D lim g(x) =3D 0, then
then l= im
f(x)
g(x)
=3D
lim
f '(x)=
g'(x)<= /span>
=
if the limit exists, or its ±∞ .
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
x= &agr= ave;a
We cal= l this type of limits as the
"= ; 0 "
0
indet= erminate
forms.=
L'Hop= ital's rule passes the calculation of the 
"= 0 "
0
form = to the calculation of derivatives.
Example: A. Find lim
sin(x)
x
xà0
Since lim sin(x) =3D 0 and lim x = =3D 0, use the L'Hopital's
rule:
xà0
xà0
lim
sin(x)
x
=3D
lim
[sin(x)]'
[x]'
xà0
xà0
=
=3D
lim
cos(x)
1
xà0
=3D 1.
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0015.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0015_image030.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhgwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACB AAEAgAAAAAAAAAIOjI+pu+APo5y02seyzgUAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0015_image031.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0015_image032.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0015_image033.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAm AAEAgAAAAAAAAAIFjI+py1YAO1== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0015_image034.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhswACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACx AAEAgAAAAAAAAAIRjI+pywcPo5y02otl27z7XgAAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0015_image035.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0002.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
B. Find lim
ex – 1
x
xà0
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0002_image036.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAn AAEAgAAAAAAAAAIFjI+py1cAO1== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0016.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
B. Find lim
ex – 1
x
xà0
Since lim ex – 1 =3D 0 and lim x =3D 0, u= se the L'Hopital's
rule:
xà0
xà0
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0016_image037.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAn AAEAgAAAAAAAAAIFjI+py1cAO1== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0017.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
B. Find lim
ex – 1
x
xà0
Since lim ex – 1 =3D 0 and lim x =3D 0, u= se the L'Hopital's
rule:
xà0
xà0
lim
ex – 1 
x
=3D
lim
[ex – 1 ]'
[x]'
xà0
xà0
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0017_image038.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAn AAEAgAAAAAAAAAIFjI+py1cAO1== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0017_image039.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhswACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACx AAEAgAAAAAAAAAIRjI+py40Ao5y02oun27x7VgAAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0018.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
B. Find lim
ex – 1
x
xà0
Since lim ex – 1 =3D 0 and lim x =3D 0, u= se the L'Hopital's
rule:
xà0
xà0
lim
ex – 1 
x
=3D
lim
[ex – 1 ]'
[x]'
xà0
xà0
=3D
lim
ex
1
xà0
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0018_image040.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAn AAEAgAAAAAAAAAIFjI+py1cAO1== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0018_image041.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhswACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACx AAEAgAAAAAAAAAIRjI+py40Ao5y02oun27x7VgAAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0018_image042.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAh AAEAgAAAAAAAAAIFjI+pmwUAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0019.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
B. Find lim
ex – 1
x
xà0
Since lim ex – 1 =3D 0 and lim x =3D 0, u= se the L'Hopital's
rule:
xà0
xà0
lim
ex – 1 
x
=3D
lim
[ex – 1 ]'
[x]'
xà0
xà0
=3D
lim
ex
1
xà0
=3D 1.
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0019_image043.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAn AAEAgAAAAAAAAAIFjI+py1cAO1== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0019_image044.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhswACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACx AAEAgAAAAAAAAAIRjI+py40Ao5y02oun27x7VgAAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0019_image045.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAh AAEAgAAAAAAAAAIFjI+pmwUAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0020.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
B. Find lim
ex – 1
x
xà0
Since lim ex – 1 =3D 0 and lim x =3D 0, u= se the L'Hopital's
rule:
xà0
xà0
lim
ex – 1 
x
=3D
lim
[ex – 1 ]'
[x]'
xà0
xà0
=3D
lim
ex
1
xà0
=3D 1.
C. Find lim
ex – 1
x2
xà0+
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0020_image046.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAn AAEAgAAAAAAAAAIFjI+py1cAO1== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0020_image047.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhswACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACx AAEAgAAAAAAAAAIRjI+py40Ao5y02oun27x7VgAAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0020_image048.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAh AAEAgAAAAAAAAAIFjI+pmwUAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0020_image049.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAm AAEAgAAAAAAAAAIFjI+py1YAO1== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0021.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
B. Find lim
ex – 1
x
xà0
Since lim ex – 1 =3D 0 and lim x =3D 0, u= se the L'Hopital's
rule:
xà0
xà0
lim
ex – 1 
x
=3D
lim
[ex – 1 ]'
[x]'
xà0
xà0
=3D
lim
ex
1
xà0
=3D 1.
C. Find lim
ex – 1
x2
xà0+
Since lim ex – 1 =3D 0 and lim x2 =3D 0, use the L'Hopital's<= span style=3D'font-size:78%;color:#006600;display:none'>
rule:
xà0+
xà0+
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0021_image050.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAn AAEAgAAAAAAAAAIFjI+py1cAO1== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0021_image051.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhswACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACx AAEAgAAAAAAAAAIRjI+py40Ao5y02oun27x7VgAAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0021_image052.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAh AAEAgAAAAAAAAAIFjI+pmwUAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0021_image053.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAm AAEAgAAAAAAAAAIFjI+py1YAO1== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0022.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
B. Find lim
ex – 1
x
xà0
Since lim ex – 1 =3D 0 and lim x =3D 0, u= se the L'Hopital's
rule:
xà0
xà0
lim
ex – 1 
x
=3D
lim
[ex – 1 ]'
[x]'
xà0
xà0
=3D
lim
ex
1
xà0
=3D 1.
C. Find lim
ex – 1
x2
xà0+
Since lim ex – 1 =3D 0 and lim x2 =3D 0, use the L'Hopital's<= span style=3D'font-size:78%;color:#006600;display:none'>
rule:
lim
ex – 1 
x2
=3D
lim
[ex – 1 ]'
[x2]'
xà0+
xà0+
xà0+
xà0+
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0022_image054.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAn AAEAgAAAAAAAAAIFjI+py1cAO1== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0022_image055.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhswACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACx AAEAgAAAAAAAAAIRjI+py40Ao5y02oun27x7VgAAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0022_image056.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAh AAEAgAAAAAAAAAIFjI+pmwUAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0022_image057.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAm AAEAgAAAAAAAAAIFjI+py1YAO1== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0022_image058.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhswACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACx AAEAgAAAAAAAAAIRjI+py30Ao5y02our27z7VQAAOw== ------=_NextPart_01C97CE5.07A1A980 Content-Location: file:///C:/0F5C3CE9/12L'Hopital'sRuleI_files/slide0023.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
B. Find lim
ex – 1
x
xà0
Since lim ex – 1 =3D 0 and lim x =3D 0, u= se the L'Hopital's
rule:
xà0
xà0
lim
ex – 1 
x
=3D
lim
[ex – 1 ]'
[x]'
xà0
xà0
=
=3D
lim
ex
1
xà0
=3D 1.
C. Find lim
ex – 1
x2
xà0+
Since lim ex – 1 =3D 0 and lim x2 =3D 0, use the L'Hopital's<= span style=3D'font-size:78%;color:#006600;display:none'>
rule:
lim
ex – 1 
x2
=3D
lim
[ex – 1 ]'
[x2]'
xà0+
=
=3D
lim
ex
2x
xà0+
xà0+
xà0+
xà0+
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
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B. Find lim
ex – 1
x
xà0
Since lim ex – 1 =3D 0 and lim x =3D 0, u= se the L'Hopital's
rule:
xà0
xà0
lim
ex – 1 
x
=3D
lim
[ex – 1 ]'
[x]'
xà0
xà0
=
=3D
lim
ex
1
xà0
=3D 1.
C. Find lim
ex – 1
x2
xà0+
Since lim ex – 1 =3D 0 and lim x2 =3D 0, use the L'Hopital's<= span style=3D'font-size:78%;color:#006600;display:none'>
rule:
lim
ex – 1 
x2
=3D
lim
[ex – 1 ]'
[x2]'
xà0+
=
=3D
lim
ex
2x
xà0+
=3D ∞.
xà0+
xà0+
xà0+
L'Hopi= tal's Rule (0/0, ∞/∞, ∞*0 forms)
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