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Cálculo I Profº. Marcello Santos Chaves 1 Marcello Santos Chaves Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 LIMITES E CONTINUIDADES ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) x x xfLim x x xfLim x x x xfLim x xfLim xx xfLim xx LimxfLim xex LimxfLim xexe eLimxfLim xexe xexLimxfLim xexe xexLimxfLim xex xex e xexLimxfLim Solução e xex xfLim e eee e ee ee ee ee ee ee e 2 )( 4 2)( 2 2 2 1)( 2 1)( 1)( 0 1 )( 1 )( )( )( )( )( : )()2 0 000 0 00 00 00 00 22 00 00 0 = ⇒=⇒⋅=⇒= × = ×+ = ×+ = ×+× = ×+× −+ = ×+× −+ = ++ ++ × −+ = −+ = → →→→ → →→ →→ →→ →→ →→ →→ → ( ) ( ) ( ) 1)( 111)( 1)( 1 1)1()( 1 1)( 1 1)( : 1 1)()1 1 2 1 2 11 2 11 3 11 3 3 3 11 3 31 = ++= ++= − ++⋅− = − − = − − = =→ − − = → → →→ →→ →→ →→ → ufLim ufLim uuLimufLim u uuuLimufLim u uLimufLim u uLimufLim xuFaça Solução x x xfLim u u uu uu uu uu x Cálculo I Profº. Marcello Santos Chaves 2 Marcello Santos Chaves Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 6)( 1 23)( 1 111111)( 1)( 1 11)( 1)( 1)( 1)( 1)( 1)( : 1)()3 1 1 2 2 1 2 2 11 2 2 11 32 3 11 22 3 11 3 11 3 11 3 3 11 3 311 −= × −= − +⋅++ = − +⋅++ = −⋅− +⋅++⋅− = − +⋅− = − +⋅− = + + ⋅ − − = − − = − − = =→ − − = → → → →→ →→ →→ →→ →→ →→ →→ →→ ufLim ufLim ufLim u uuuuuLimufLim uu uuuuuuLimufLim uu uuuuLimufLim uuu uuuuLimufLim uuu uuu uuu uLimufLim uuu uLimufLim uu uLimufLim uxFaça Solução xx xLimxfLim u u u uu uu uu uu uu uu uu xx ( ) ( ) ( ) ( ) 1)( 111 1)( 1 1)( 11 1)( 1 1)( 11 : 11)()4 0 20 200 200 300 33 3 00 = ++ = ++ = ++⋅− − = − − = −=⇔+=→ −+ = → → →→ →→ →→ →→ ufLim ufLim uu LimufLim uuu uLimufLim u uLimufLim uxxuFaça Solução x xLimxfLim u u uu uu uu xx Cálculo I Profº. Marcello Santos Chaves 3 Marcello Santos Chaves Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 ( ) ( )( ) ( ) ( ) ( ) ( ) 6 3)( 34 32)( 32 32 32 1)( 33 1)( 330 1)( 33 1)( 33 )( 33 33)( 33 33)( 33 3333)( : 33)()5 0 00 0 0 00 00 00 22 00 00 00 = × =⇒⋅= + = ++ = ++ = ++⋅ = ++⋅ −+ = ++⋅ −+ = ++ ++ ⋅ −+ = −+ = → →→ → → →→ →→ →→ →→ →→ →→ xfLim xfLimxfLim xfLim xfLim x LimxfLim xx xLimxfLim xx xLimxfLim xx xLimxfLim x x x xLimxfLim Solução x xLimxfLim x xx x x xx xx xx xx xx xx ( )[ ] ( ) −∞= ∞− = ⋅+ ∞+⋅−⋅ = ⋅+ −⋅ = +⋅ −⋅ = + − = + − = +∞→ +∞→ +∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ )( 8 )( 028 305)( 128 315 )( 28 35 )( 28 35)( : 28 35)()6 2 2 2 3 3 xfLim xfLim xfLim x x xLimxfLim x x x x x LimxfLim x xLimxfLim Solução x xLimxfLim x x x xx xx xx xx Cálculo I Profº. Marcello Santos Chaves 4 Marcello Santos Chaves Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 ( ) ( ) 7 5)( 037 025)( 137 125 )( 37 25 )( 37 25)( : 37 25)()7 3 3 3 3 3 3 3 3 3 3 − = ⋅+ ⋅−− = ⋅+ ⋅−− = +⋅ −⋅− = + +− = + +− = +∞→ +∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ xfLim xfLim x xLimxfLim x x x x LimxfLim x xLimxfLim Solução x xLimxfLim x x xx xx xx xx 1)( 1 1)( 01 01)( 11 11 )( 11 11 )( 11 11 )( 11 11 )( 1 1)( : 1 1)()8 2 2 2 2 2 2 2 2 = = + + = + + = +⋅ +⋅ = +⋅ +⋅ = +⋅ +⋅ = + + = + + = +∞→ +∞→ +∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ xfLim xfLim xfLim x x LimxfLim x x x x LimxfLim x x x x LimxfLim x x x x LimxfLim x xLimxfLim Solução x xLimxfLim x x x xx xx xx xx xx xx Cálculo I Profº. Marcello Santos Chaves 5 Marcello Santos Chaves Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 ( ) ( ) ( ) ( ) ( ) 0)( 2 0)( 11 0)( 0101 02)( 1111 12 )( 1111 2 )( 1111 2)( 1111 11)( 1111 11)( 1111 11)( 11 1111)( : 11)()9 22 22 22 22 22 2 2 2 2 22 2 2 2 2 2 2 2 2 22 22 22 22 = = + = −++ ⋅ = −++ ⋅ = −++⋅ = −++⋅ = −⋅+ +⋅ +−+ = −⋅++⋅ −−+ = −⋅+ +⋅ −−+ = −++ −++ ⋅−−+= −−+= +∞→ +∞→ +∞→ +∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ xfLim xfLim xfLim xfLim xx xLimxfLim x xx x xLimxfLim xx x LimxfLim x x x x xxLimxfLim x x x x xxLimxfLim x x x x xxLimxfLim xx xx xxLimxfLim Solução xxLimxfLim x x x x xx xx xx xx xx xx xx xx Cálculo I Profº. Marcello Santos Chaves 6 Marcello Santos Chaves Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 ( ) ( ) ( ) ( )[ ] ( )( ) ( )[ ] ( )( ) ( ) 2 1)( 11 1)( 101 1)( 111 1)( 111 1)( 1 11 1)( 1 11 1)( 11 1)( 11 )( 11 )( 111)( 1 1)( 1 11)( 1)( 1)( : 1)()10 2 22 2 2 2 2 2 2 2 2 2 424 2 2 422 22 22 2 2 22 22 22 22 2 2 −= ⇒ + − =⇒ +− − =⇒ +− − =⇒ ⇒ + −⋅ − =⇒ + −⋅ − =⇒ ⇒ + −⋅ − =⇒ + − − = + − ⋅ − = + − ⋅ −− = + − ⋅ −⋅− = +⋅− −⋅− = +⋅− +⋅− ⋅−⋅−= −⋅−= −−= −−= +∞→ +∞→+∞→+∞→+∞→ +∞→+∞→+∞→+∞→ +∞→+∞→+∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ xfLim xfLimxfLim x LimxfLim x x x LimxfLim x x x LimxfLim x x x LimxfLim x x LimxfLim x x x xLimxfLim x x x xxxLimxfLim x x x xxxLimxfLim xxx xxxLimxfLim xxx xxx xxxLimxfLim xxxLimxfLim xxxLimxfLim Solução xxxLimxfLim x xxxx xxxx xxxx xx xx xx xx xx xx xx xx Cálculo I Profº. Marcello Santos Chaves 7 Marcello Santos Chaves Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 +∞= ∞+ = − −∞+ = − − = −⋅ −⋅ = − − = − −⋅ = − − = =→ − − = +∞→ +∞→ +∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ )( 3 )( 03 0)( 13 1 )( 13 1 )( 13 1)( 13 1)( 13 1)( : 13 1)()11 2 2 2 2 2 2 2 3 2 2 2 22 2 xfLim xfLim xfLim x x x LimxfLim x x x xx LimxfLim x xLimxfLim x xxLimxfLim x xxLimxfLim xvFaça Solução v vvLimvfLim x x x xx xx xx xx xx vv +∞= ∞+ = − −∞+ = − − = −⋅ −⋅ = − − = − −⋅ = − − = =→ − − = +∞→ +∞→ +∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ +∞→+∞→ )( 3 )( 03 0)( 13 1 )( 13 1 )( 13 1)( 13 1)( 13 1)( : 13 1)()11 2 2 2 2 2 2 2 3 2 2 2 22 2 xfLim xfLim xfLim x x x LimxfLim x x x xx LimxfLim x xLimxfLim x xxLimxfLim x xxLimxfLim xvFaça Solução v vvLimvfLim x x x xx xx xx xx xx vv Cálculo I Profº. Marcello Santos Chaves 8 Marcello Santos Chaves Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 8)( 12 32)( 444 148)( 4222 12242)( 42 124)( 422 1224)( 222 124)( 2 144)( 8 14)( 1 1 8 16)( 8 16)( : 8 16)()12 2 2 2 2 2 2 2 2 22 2 2 22 22 222 22 33 22 22 3 222 22 3 4 22 3 4 22 3 4 22 −= − = ++ −×× = +⋅+ −⋅+⋅+ = ++ −⋅+⋅+ = ++⋅− −⋅−⋅+⋅+ = ++⋅− −⋅−⋅+ = − −⋅−⋅+ = − −⋅− = − − ⋅ − − = − − = − − = → → → → →→ →→ →→ →→ →→ →→ →→ →→ xfLim xfLim xfLim xfLim xx xxLimxfLim xxx xxxLimxfLim xxx xxLimxfLim x xxLimxfLim x xLimxfLim x xLimxfLim x xLimxfLim Solução x xLimxfLim x x x x xx xx xx xx xx xx xx xx ( ) ( ) ( ) ( ) 3 2)( 111 11)( 1 1)( 111 11)( 1 1)( 1 1)( 1 1)( : 1 1)()13 1 21 211 2211 33 22 11 3 2 11 6 3 6 11 6 3 11 = ++ + = ++ + = +⋅+⋅− −⋅+ = − − = − − = − − = =→ − − = → → →→ →→ →→ →→ →→ →→ tfLim tfLim tt tLimtfLim ttt ttLimtfLim t tLimtfLim t tLimtfLim t tLimtfLim txFaça Solução x xLimxfLim t t tt tt tt tt tt xx
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