Limites de Sequências:
Conceito e exemplos intuitivos. Definição de limite de uma sequência
Sequências Convergentes e Divergentes. Exemplos.
Definição de número Neperiano “e” como limite de uma sequência
Critério da Comparação para Convergência (Teorema do Sanduíche)
Equivalência de limite por epsilon-delta e por sequências (importante para dar contraexemplos da não existência de limite)
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Exercício 3 - Vídeo aula 8
Calcule:
![\displaystyle\lim_{n\to \infty}\sqrt[n]{3^n+5^n}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ukwSw1MHtSBahGHAspx2ggtwoiNlQ73HGi_86RBPBrTjj0ZZdFIPt8FqfHzTejAuX8cs4BrKetsDTZ-tGXeMKOce9dQXICLKf611tYXIlt8YmgRuy9AnjpqP2mCUiXvpJG2yT8Fz_hzoBjsgtQRZg3PlMzYUnFSgLkzSwnMeGj7WrB-ruZas6xB5yVYPk2yDUOvif7X3Eb2N1IUk6AWBMG21l-pG_CHvA=s0-d "\displaystyle\lim_{n\to \infty}\sqrt[n]{3^n+5^n}")
![\displaystyle\lim_{n\to \infty}\sqrt[n]{3^n+5^n}= \displaystyle\lim_{n\to \infty} e^{\mathrm{ln}(3^n+5^n)^{\frac{1}{n}}}=\displaystyle\lim_{n\to \infty} e^{\mathrm{ln}5\cdot((\frac{3}{5})^n+1)^{\frac{1}{n}}}\Rightarrow](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uRLEZa4hFvmnnGAZ_vEJ55x2YyG9VXKOYr47w0ADGdLd_c_SnkGRz6XseK1SFmXRUw9yzRt4IC8uZmXtUu9PHCULhLojk2Gr-gqk67Fj0J2OM3V0gEsfB0e8htZEP3ZLYzBH71fPEKHVV5hEKSRokXTat0-1FaHFWXjr3ieSDdcjmUWjwQAexG81k6BTSG-1rAtj_adHd_6hApQme1S_Vh1mSMt_O0965sSzYZRRkz9Y0C8lBJqa5ub7MKRrw_YUy03nP6fWgzfUAdN7mjmou7mdybYef4vOsJexsUaPxeZtyDfy26QRyd4s-7lpqlXqyRiowXFFEkVYI7_FQLyhZY4BF-IixkI2yDdwhw3F50GEk2FHMNXMk0EprPNdrHdu_9UoWj9kCkSe3blJAnzYsdfnEmGPu643gYaj2uW-J3urGYjgX_nVF3aoh33AkYSoQh072676cFHVlfKDIvr230rUT6KxpXWpMnXgVR60vXTOfaOfPDy25a2jtwDmmLASS5rjqoyHbAGC5f3ArmRZLQFYV2trEoWwUTnrEuzp6AfJvZmhprxoL9j3KUljO3hsM6mgtD8s2qCHIYCg8=s0-d "\displaystyle\lim_{n\to \infty}\sqrt[n]{3^n+5^n}= \displaystyle\lim_{n\to \infty} e^{\mathrm{ln}(3^n+5^n)^{\frac{1}{n}}}=\displaystyle\lim_{n\to \infty} e^{\mathrm{ln}5\cdot((\frac{3}{5})^n+1)^{\frac{1}{n}}}\Rightarrow")
![\displaystyle\lim_{n\to \infty}\sqrt[n]{3^n+5^n}= \displaystyle\lim_{n\to \infty} e^{\mathrm{ln}5}\cdot \displaystyle\lim_{n\to \infty}e^{\frac{\mathrm{ln}((\frac{3}{5})^n+1)}{n}}=5\cdot e^{\displaystyle\lim_{n\to \infty}\frac{\mathrm{ln}((\frac{3}{5})^n+1)}{n}}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_toDIbZerkw9xr0ApPB7iPE-Bff2LEUAX-_Hr4hjOil2b96wnYF--tYOijqvHvnEex91mYh-W8QMXU7pMM5LNLnJvRxAqvMeJxF1uzHRotR6q-EWmBEcKlH0fi_POBzX1dyVmNBteKhDvFzJwcphtC6NzaabpYqKA_O_feq_G9ncHKD31AWKP_gUMFzRzsLWCkEJ5vLVTknlr5YmOh09CUWbmIngKZe63DmF76TH_CD3HvkalXmcQFEYlLkuVjQf9VCsyyEbYYguRZhljmfjZ6x1CO0SLz705wpbZ9BI0mu6SQKI9qr1Ox8Kxnf41CR-Hv9BpBs3w-BkQhW2i1e6zqaU8xcDoraChTyIOlg3VUKp4MVh2yZ-xQRBDJclzjL0R-azNf_cS4tXQ9g-BGltSib403wHKeqnIy5HzqahmYwFgGIexjoD_stzDe5akZed6aiJWwC9uVAv2U_ICyjc6Ozt2MHlewaSZ3orU5fA3QBL3gi0fiNf9Jp0IGybH1B6LX9Lup-h6IAuexNEphbklKfT8mgLdA62nG6lIqDdrAiSFGH0bsD62hQrcFPhT54XScgpCr3VAieGGdtV64qak6eFKMr177HdLKoxIibR4G6Jdzk0P0V7f2NooobrXd6rwy0_ee6WQyxY4FLiylXAYVn9Ze9HGzd-1zS21UW4Q=s0-d "\displaystyle\lim_{n\to \infty}\sqrt[n]{3^n+5^n}= \displaystyle\lim_{n\to \infty} e^{\mathrm{ln}5}\cdot \displaystyle\lim_{n\to \infty}e^{\frac{\mathrm{ln}((\frac{3}{5})^n+1)}{n}}=5\cdot e^{\displaystyle\lim_{n\to \infty}\frac{\mathrm{ln}((\frac{3}{5})^n+1)}{n}}")
. Como
![\displaystyle\lim_{n\to \infty}\left(\dfrac{3}{5}\right)^n=0 \Rightarrow \displaystyle\lim_{n\to \infty}\sqrt[n]{3^n+5^n} = 5\cdot e^{\displaystyle\lim_{n\to \infty}\frac{\mathrm{ln}(0+1)}{n}}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tPGdFXnxoIM7ONeLpusbehFVlSaWtrwMRUNj_PLHkI3L1fGRGTy-_fMELGBzcYXi2mnMrvg-YGE6MJKA2-zAjYf9DuMyyPSSGvzsPHpvf90xkgEYeSuDAsKLGcNkRAjXKUL8GNOMAOIZSmCCJuAogirkUWRnDOPApuzSuGm5EXR3SFws24hXNksSkbHODamaDitZplOASOV1LOEpZKF2rsgngXXYpb_hYQiqxyeYUZTdMOXMS27_tbsxeAVpQ9Fsv4H5hA-pbYGq6OevsSDEHqZavt4IddWqOkiqa-0FSEibYMTgpgoOb_m3358v0CQ_8Oq09uS5rk-8Q6HiL7wiSotp7FDP4Q8gIRPv8FCJrpKE9wOHl3B5Mp5LbSAwHgoKxihgX-Rmk9mGo2RqNkhquC8XkwxG2H01xs5b2-VAtIbAPQr5A38hM73tibD1aI24MNbK1TXgBLwt_cbt8THp5ZsvdNcdkYvszs8-1ZoHfhKPFg9vGZytGhw4zQuj042w=s0-d "\displaystyle\lim_{n\to \infty}\left(\dfrac{3}{5}\right)^n=0 \Rightarrow \displaystyle\lim_{n\to \infty}\sqrt[n]{3^n+5^n} = 5\cdot e^{\displaystyle\lim_{n\to \infty}\frac{\mathrm{ln}(0+1)}{n}}")
Portanto,
![\displaystyle\lim_{n\to \infty}\sqrt[n]{3^n+5^n}=5\cdot e^0=5](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uR1VwxkBbSvmNClk80a1fHh_T-kVggxMoDh3SZzUjhUv1BhNXcoiMu4Nz01yk-XkpSxyyi9omkJ5b8-kdR0twhP0RrOgikQC7Xf01JhJDP5UEMqIK7BovQ7zIOsV_DNtFf08h-CCQjFlaDE3ksTTdHqKBSl9or7Rzyoge1SjxVGgWdGfiX1NSM3dvXYA9KK8hLyRJ90Di3kBxxmpbQ3PEaujZ-uu0k_cmvqrkqkivayGXr_69qyeHHpF7a8IM=s0-d "\displaystyle\lim_{n\to \infty}\sqrt[n]{3^n+5^n}=5\cdot e^0=5")
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