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L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0003.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Theore= m: Let k > 0, then lim k1/x =3D 1.
x&agr= ave;∞
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0004.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Theore= m: Let k > 0, then lim k1/x =3D 1.
In pic= ture:
x&agr= ave;∞
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
0
1
K
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Theore= m: Let k > 0, then lim k1/x =3D 1.
In pic= ture:
x&agr= ave;∞
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
0
1
K
K1/2
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0053_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhqQECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACn AQIAgAAAAAAAAAIbjI+py+0Po5y02ouz3rz7D4biSJbmiabqynoFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0053_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAC ABIAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0053_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAJ ABIAgQAAAAAAALvg4wECAwIhhBGnuJfNWphqikvR3cJyHHyg2IWi9yVoV01PBUVS/AIFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0053_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAP AA4AgQAAAAAAAOEJDgECAwIlhB2pxykPAwMB2ifR3blu230gJV5OGZFouB6emKlfjLwY3Sh4AQA7 ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0053_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAK AAoAgQAAAAAAAOEJDgECAwIahIOmEOISVHsuTgoZfmjXS0UaJnbcwiAiUAAAOw== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0054.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Theore= m: Let k > 0, then lim k1/x =3D 1.
In pic= ture:
x&agr= ave;∞
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
0
1
K
K1/2
K1/3
K1/4
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0054_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhqQECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACn AQIAgAAAAAAAAAIbjI+py+0Po5y02ouz3rz7D4biSJbmiabqynoFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0054_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAC ABIAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0054_image012.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAJ ABIAgQAAAAAAALvg4wECAwIhhBGnuJfNWphqikvR3cJyHHyg2IWi9yVoV01PBUVS/AIFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0054_image013.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAP AA4AgQAAAAAAAOEJDgECAwIlhB2pxykPAwMB2ifR3blu230gJV5OGZFouB6emKlfjLwY3Sh4AQA7 ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0054_image014.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOAALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA2 AAoAgQAAAAAAAOEJDgECAwJLhIOmyw0YnFSQBYFFnA7lxFEXBo6ZFlonmT6rZp5bG39pTeIs/aKh 7unNfsKbENg6FImvCG7IXEE7xeAu+YhNJVbfoYJdQLacsaEAADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0054_image015.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhbAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABABm AA0AgAAAAOEJDgJDhI+py+0PXwix2ovzmlP7D14cF5ZmOY5JyrbuC8fyTKtGjef6jgP8DwwKh8Qi 7YRMOl7KpvPQekqbqak1SbpqTZRHAQA7 ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0055.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Theore= m: Let k > 0, then lim k1/x =3D 1.
In pic= ture:
x&agr= ave;∞
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
0
1
K
K
K1/2
K1/3
K1/4
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0055_image016.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhqQECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACn AQIAgAAAAAAAAAIbjI+py+0Po5y02ouz3rz7D4biSJbmiabqynoFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0055_image017.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAC ABIAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0055_image018.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAJ ABIAgQAAAAAAALvg4wECAwIhhBGnuJfNWphqikvR3cJyHHyg2IWi9yVoV01PBUVS/AIFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0055_image019.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAO AA4AgQAAAAAAAOEJDgECAwIjhBGpxyEPU4P0BeMqvUjX5H1ZaI0kSFpdepkat74w4s6NMhcAOx== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0055_image020.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAP AA4AgQAAAAAAAOEJDgECAwIlhB2pxykPAwMB2ifR3blu230gJV5OGZFouB6emKlfjLwY3Sh4AQA7 ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0055_image021.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOAALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA2 AAoAgQAAAAAAAOEJDgECAwJLhIOmyw0YnFSQBYFFnA7lxFEXBo6ZFlonmT6rZp5bG39pTeIs/aKh 7unNfsKbENg6FImvCG7IXEE7xeAu+YhNJVbfoYJdQLacsaEAADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0055_image022.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhbAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABABm AA0AgAAAAOEJDgJDhI+py+0PXwix2ovzmlP7D14cF5ZmOY5JyrbuC8fyTKtGjef6jgP8DwwKh8Qi 7YRMOl7KpvPQekqbqak1SbpqTZRHAQA7 ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0056.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Theore= m: Let k > 0, then lim k1/x =3D 1.
In pic= ture:
x&agr= ave;∞
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
0
1
K
K1/2
K
K1/2
K1/3
K1/4
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0056_image023.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhqQECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACn AQIAgAAAAAAAAAIbjI+py+0Po5y02ouz3rz7D4biSJbmiabqynoFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0056_image024.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAC ABIAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0056_image025.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAJ ABIAgQAAAAAAALvg4wECAwIhhBGnuJfNWphqikvR3cJyHHyg2IWi9yVoV01PBUVS/AIFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0056_image026.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAO AA4AgQAAAAAAAOEJDgECAwIjhBGpxyEPU4P0BeMqvUjX5H1ZaI0kSFpdepkat74w4s6NMhcAOx== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0056_image027.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAJ AAoAgQAAAAAAAOEJDgECAwIVhIN2IZ3cYgAwumqRlZhvR23TgowFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0056_image028.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAP AA4AgQAAAAAAAOEJDgECAwIlhB2pxykPAwMB2ifR3blu230gJV5OGZFouB6emKlfjLwY3Sh4AQA7 ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0056_image029.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOAALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA2 AAoAgQAAAAAAAOEJDgECAwJLhIOmyw0YnFSQBYFFnA7lxFEXBo6ZFlonmT6rZp5bG39pTeIs/aKh 7unNfsKbENg6FImvCG7IXEE7xeAu+YhNJVbfoYJdQLacsaEAADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0056_image030.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhbAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABABm AA0AgAAAAOEJDgJDhI+py+0PXwix2ovzmlP7D14cF5ZmOY5JyrbuC8fyTKtGjef6jgP8DwwKh8Qi 7YRMOl7KpvPQekqbqak1SbpqTZRHAQA7 ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0057.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Theore= m: Let k > 0, then lim k1/x =3D 1.
In pic= ture:
x&agr= ave;∞
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
0
1
K
K1/2
K1/3
K1/4
K
K1/2
K1/3
K1/4
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Theore= m: Let k > 0, then lim k1/x =3D 1.
In pic= ture:
x&agr= ave;∞
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
0
1
K
K1/2
K1/3
K1/4
K
K1/2
K1/3
K1/4
But instead of having a constant base k, what wil= l
happen if k changes?
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Theore= m: Let k > 0, then lim k1/x =3D 1.
In pic= ture:
x&agr= ave;∞
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
0
1
K
K1/2
K1/3
K1/4
K
K1/2
K1/3
K1/4
But instead of having a constant base k, what wil= l
happen if k changes?
In ge= neral, if b(x) à ∞ and e(x) à 0 as x à a, what = happens to b(x)e(x) as x = à a?
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Theore= m: Let k > 0, then lim k1/x =3D 1.
In pic= ture:
x&agr= ave;∞
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
0
1
K
K1/2
K1/3
K1/4
K
K1/2
K1/3
K1/4
But instead of having a constant base k, what wil= l
happen if k changes?
In ge= neral, if b(x) à ∞ and e(x) à 0 as x à a, what happens to b(x)e(x) as x = à a?  We say t= hese limit problems are of the forms ∞0.
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0007_image052.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhqQECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACn AQIAgAAAAAAAAAIbjI+py+0Po5y02ouz3rz7D4biSJbmiabqynoFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0007_image053.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAC ABIAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0007_image054.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAJ ABIAgQAAAAAAALvg4wECAwIhhBGnuJfNWphqikvR3cJyHHyg2IWi9yVoV01PBUVS/AIFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0007_image055.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAO AA4AgQAAAAAAAOEJDgECAwIjhBGpxyEPU4P0BeMqvUjX5H1ZaI0kSFpdepkat74w4s6NMhcAOx== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0007_image056.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAJ AAoAgQAAAAAAAOEJDgECAwIVhIN2IZ3cYgAwumqRlZhvR23TgowFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0007_image057.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKwEUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAn AQ8AgQAAAAAAAOEJDgECAwLshBGpx+0Po5y02ouz3rz7byXCSAbgiabqyrbuG5DyaL72jef6zsbz XOMJh5AR8Yi8+H4l24LzxESTsociOLkmt6clU4BdLRMZES0cMYPRu5/Dy27AuXTOF9jy0qS/uLw/ 9PWHlwZYd2hxJ+PXobcXwsRo4AjGo3iJmam5ydnp+YlJuZj3JXlAaYqqA8ra6voK+yra1FMKaSih mhPL2+v7+4mgaLqhW2FciLsLzNzsHCts6+JIPBlJgbzKNDia3I0InqVHBjMnZY6NHjjztiipHh4P cVUNolU8dZxPVPW2P/9PnsCBBFMYKYjwRAEAOw== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0007_image058.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhbAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABABm AA0AgAAAAOEJDgJDhI+py+0PXwix2ovzmlP7D14cF5ZmOY5JyrbuC8fyTKtGjef6jgP8DwwKh8Qi 7YRMOl7KpvPQekqbqak1SbpqTZRHAQA7 ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0008.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Theore= m: Let k > 0, then lim k1/x =3D 1.
In pic= ture:
x&agr= ave;∞
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
0
1
K
K1/2
K1/3
K1/4
K
K1/2
K1/3
K1/4
But instead of having a constant base k, what wil= l
happen if k changes?
In ge= neral, if b(x) à ∞ and e(x) à 0 as x à a, what happens to b(x)e(x) as x = à a?  We say t= hese limit problems are of the forms ∞0.
Simil= arly, if b(x) à 0 and e(x) à 0 as x à a, we say these problems "lim b(x)e(x)" are of the forms 00.
x&agr= ave;a
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Theore= m: Let k > 0, then lim k1/x =3D 1.
In pic= ture:
x&agr= ave;∞
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
0
1
K
K1/2
K1/3
K1/4
K
K1/2
K1/3
K1/4
But instead of having a constant base k, what wil= l
happen if k changes?
In general, if b(x) à = ∞ and e(x) à 0 as x &agrav= e; a, what happens to b(x)e(x) as x = à a?  We say t= hese limit problems are of the forms ∞0.
Similarly, if b(x) à 0 and e(x) à 0 as x à a, we say these problems "lim b(x)e(x)" are of the forms 00.
x&agr= ave;a
Lastly, we have the 1∞ forms, i.e. the problems of
"lim b(= x)e(x)" where b(x) à 1 and e(x) = à ∞.
x= &agr= ave;a
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0009_image066.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhqQECAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACn AQIAgAAAAAAAAAIbjI+py+0Po5y02ouz3rz7D4biSJbmiabqynoFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0009_image067.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAC ABIAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0009_image068.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAJ ABIAgQAAAAAAALvg4wECAwIhhBGnuJfNWphqikvR3cJyHHyg2IWi9yVoV01PBUVS/AIFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0009_image069.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAO AA4AgQAAAAAAAOEJDgECAwIjhBGpxyEPU4P0BeMqvUjX5H1ZaI0kSFpdepkat74w4s6NMhcAOx== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0009_image070.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAJ AAoAgQAAAAAAAOEJDgECAwIVhIN2IZ3cYgAwumqRlZhvR23TgowFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0009_image071.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKwEUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAn AQ8AgQAAAAAAAOEJDgECAwLshBGpx+0Po5y02ouz3rz7byXCSAbgiabqyrbuG5DyaL72jef6zsbz XOMJh5AR8Yi8+H4l24LzxESTsociOLkmt6clU4BdLRMZES0cMYPRu5/Dy27AuXTOF9jy0qS/uLw/ 9PWHlwZYd2hxJ+PXobcXwsRo4AjGo3iJmam5ydnp+YlJuZj3JXlAaYqqA8ra6voK+yra1FMKaSih mhPL2+v7+4mgaLqhW2FciLsLzNzsHCts6+JIPBlJgbzKNDia3I0InqVHBjMnZY6NHjjztiipHh4P cVUNolU8dZxPVPW2P/9PnsCBBFMYKYjwRAEAOw== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0009_image072.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhbAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABABm AA0AgAAAAOEJDgJDhI+py+0PXwix2ovzmlP7D14cF5ZmOY5JyrbuC8fyTKtGjef6jgP8DwwKh8Qi 7YRMOl7KpvPQekqbqak1SbpqTZRHAQA7 ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0010.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hop= ital's Rule II (∞0, 0<= /span>0<= /span>, 1∞, ∞ – ∞  forms)
These are the three indeterminate forms related t= o
limits of power functions b(x)e= (x) . <= /span>
(∞0, 0<= /span>0= , 1∞)
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0011.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
These are the three indeterminate forms related t= o
limits of power functions b(x)e= (x) . <= /span>
We find their limits by first finding the limits of their = logs i.e. Ln(b(x)e(x)) =3D e(x)*<= /span>Ln(b(x)).
(∞0, 0<= /span>0= , 1∞)
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0012.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
These are the three indeterminate forms related t= o
limits of power functions b(x)e= (x) . <= /span>
We find their limits by first finding the limits of their = logs i.e. Ln(b(x)e(x)) =3D e(x)*<= /span>Ln(b(x)).
Example: A. Find lim x1/x. &= #13;
xà∞ <= /div>
(∞0, 0<= /span>0= , 1∞)
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0013.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
These are the three indeterminate forms related t= o
limits of power functions b(x)e= (x) . <= /span>
We find their limits by first finding the limits of their = logs i.e. Ln(b(x)e(x)) =3D e(x)*<= /span>Ln(b(x)).
Example: A. Find lim x1/x. ( W= 34;0 
xà∞ <= /div>
(∞0, 0<= /span>0= , 1∞)
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0014.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
These are the three indeterminate forms related t= o
limits of power functions b(x)e= (x) . <= /span>
We find their limits by first finding the limits of their = logs i.e. Ln(b(x)e(x)) =3D e(x)*<= /span>Ln(b(x)).
Example: A. Find lim x1/x. ( W= 34;0 
xà∞ <= /div>
We find the limit of Ln(x1/x) =3D        = ;  first.
Ln(x)
x
(∞0, 0<= /span>0= , 1∞)
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0014_image073.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAt AAEAgAAAAAAAAAIFjI+py10AO1== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0015.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
These are the three indeterminate forms related t= o
limits of power functions b(x)e= (x) . <= /span>
We find their limits by first finding the limits of their = logs i.e. Ln(b(x)e(x)) =3D e(x)*<= /span>Ln(b(x)).
Example: A. Find lim x1/x. ( W= 34;0 
xà∞ <= /div>
lim Ln(x1/x) =3D  =
We find the limit of Ln(x1/x) =3D        = ;  first.
Ln(x)
x
lim
Ln(x)
x
xà∞ <= /div>
xà∞ <= /div>
(∞0, 0<= /span>0= , 1∞)
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0015_image074.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAt AAEAgAAAAAAAAAIFjI+py10AO1== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0015_image075.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAt AAEAgAAAAAAAAAIFjI+py10AO1== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0016.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
These are the three indeterminate forms related t= o
limits of power functions b(x)e= (x) . <= /span>
We find their limits by first finding the limits of their = logs i.e. Ln(b(x)e(x)) =3D e(x)*<= /span>Ln(b(x)).
Example: A. Find lim x1/x. ( W= 34;0 
xà∞ <= /div>
lim Ln(x1/x) =3D  =
We find the limit of Ln(x1/x) =3D        = ;  first.
Ln(x)
x
lim
Ln(x)
x
xà∞ <= /div>
xà∞ <= /div>
this is W= 34;/∞ form so use the L'Hopital's Rule.=
(∞0, 0<= /span>0= , 1∞)
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0016_image076.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAt AAEAgAAAAAAAAAIFjI+py10AO1== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0016_image077.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAt AAEAgAAAAAAAAAIFjI+py10AO1== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0016_image078.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIwAMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAf AAcAgAAAAAAAAAIYhB+ny42RnHSw2ouz3hybO4VNJZZLZDoFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0017.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
These are the three indeterminate forms related t= o
limits of power functions b(x)e= (x) . <= /span>
We find their limits by first finding the limits of their = logs i.e. Ln(b(x)e(x)) =3D e(x)*<= /span>Ln(b(x)).
Example: A. Find lim x1/x. ( W= 34;0 
xà∞ <= /div>
lim Ln(x1/x) =3D  =
We find the limit of Ln(x1/x) =3D        = ;  first.
Ln(x)
x
lim
Ln(x)
x
xà∞ <= /div>
xà∞ <= /div>
=3D 
this is W= 34;/∞ form so use the L'Hopital's Rule.=
lim
[Ln(x)]'
[x]'
xà∞ <= /div>
(∞0, 0<= /span>0= , 1∞)
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0017_image079.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAt AAEAgAAAAAAAAAIFjI+py10AO1== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0017_image080.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAt AAEAgAAAAAAAAAIFjI+py10AO1== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0017_image081.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAt AAEAgAAAAAAAAAIFjI+py10AO1== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0017_image082.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIwAMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAf AAcAgAAAAAAAAAIYhB+ny42RnHSw2ouz3hybO4VNJZZLZDoFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0018.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
These are the three indeterminate forms related t= o
limits of power functions b(x)e= (x) . <= /span>
We find their limits by first finding the limits of their = logs i.e. Ln(b(x)e(x)) =3D e(x)*<= /span>Ln(b(x)).
Example: A. Find lim x1/x. ( W= 34;0 
xà∞ <= /div>
lim Ln(x1/x) =3D  =
We find the limit of Ln(x1/x) =3D        = ;  first.
Ln(x)
x
lim
Ln(x)
x
xà∞ <= /div>
xà∞ <= /div>
=3D 
this is W= 34;/∞ form so use the L'Hopital's Rule.=
lim
[Ln(x)]'
[x]'
xà∞ <= /div>
=3D 
lim
1
x
xà∞ <= /div>
(∞0, 0<= /span>0= , 1∞)
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0018_image083.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAt AAEAgAAAAAAAAAIFjI+py10AO1== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0018_image084.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAt AAEAgAAAAAAAAAIFjI+py10AO1== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0018_image085.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAt AAEAgAAAAAAAAAIFjI+py10AO1== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0018_image086.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0018_image087.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIwAMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAf AAcAgAAAAAAAAAIYhB+ny42RnHSw2ouz3hybO4VNJZZLZDoFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0019.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
These are the three indeterminate forms related t= o
limits of power functions b(x)e= (x) . <= /span>
We find their limits by first finding the limits of their = logs i.e. Ln(b(x)e(x)) =3D e(x)*<= /span>Ln(b(x)).
Example: A. Find lim x1/x. ( W= 34;0 
xà∞ <= /div>
lim Ln(x1/x) =3D  =
We find the limit of Ln(x1/x) =3D        = ;  first.
Ln(x)
x
lim
Ln(x)
x
xà∞ <= /div>
xà∞ <= /div>
=3D 
this is W= 34;/∞ form so use the L'Hopital's Rule.=
lim
[Ln(x)]'
[x]'
xà∞ <= /div>
=3D 
lim
1
x
xà∞ <= /div>
=3D 0. 
(∞0, 0<= /span>0= , 1∞)
------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0019_image088.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAt AAEAgAAAAAAAAAIFjI+py10AO1== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0019_image089.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAt AAEAgAAAAAAAAAIFjI+py10AO1== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0019_image090.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAt AAEAgAAAAAAAAAIFjI+py10AO1== ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0019_image091.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0019_image092.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIwAMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAf AAcAgAAAAAAAAAIYhB+ny42RnHSw2ouz3hybO4VNJZZLZDoFADs= ------=_NextPart_01C97CE5.2D2EB300 Content-Location: file:///C:/70F31D69/13L'Hopital'sRuleII_files/slide0020.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
These are the three indeterminate forms related t= o
limits of power functions b(x)e= (x) . <= /span>
We find their limits by first finding the limits of their = logs i.e. Ln(b(x)e(x)) =3D e(x)*<= /span>Ln(b(x)).
Example: A. Find lim x1/x. ( W= 34;0 
xà∞ <= /div>
lim Ln(x1/x) =3D  =
We find the limit of Ln(x1/x) =3D        = ;  first.
Ln(x)
x
lim
Ln(x)
x
xà∞ <= /div>
xà∞ <= /div>
=3D 
this is W= 34;/∞ form so use the L'Hopital's Rule.=
lim
[Ln(x)]'
[x]'
xà∞ <= /div>
=3D 
lim
1
x
xà∞ <= /div>
=3D 0. 
So lim x1/x =3D
xà∞ <= /div>
lim
eLn(x    )
1/x
(∞0, 0<= /span>0= , 1∞)
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L'Hop= ital's Rule II (∞0, 0<= /span>0= , 1∞, ∞ – ∞  forms)
These are the three indeterminate forms related t= o
limits of power functions b(x)e= (x) . <= /span>
We find their limits by first finding the limits of their = logs i.e. Ln(b(x)e(x)) =3D e(x)*<= /span>Ln(b(x)).
Example: A. Find lim x1/x. ( W= 34;0 
xà∞ <= /div>
lim Ln(x1/x) =3D 
We find the limit of Ln(x1/x) =3D        = ;  first.
Ln(x)
x
lim
Ln(x)
x
xà∞ <= /div>
xà∞ <= /div>
=3D 
this is W= 34;/∞ form so use the L'Hopital's Rule.=
lim
[Ln(x)]'
[x]'
xà∞ <= /div>
=3D 
lim
1
x
xà