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anonymous

  • one year ago

Evaluate. \[\lim_{x \rightarrow \infty}\left[ \log_{5}(\frac{ 1 }{ 125 }-2^{-x}) \right]\]

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  1. rational
    • one year ago
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    log is continuous function, so you can send the limit inside the function : \[\lim(\log(f(x))) = \log(\lim f(x))\]

  2. rational
    • one year ago
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    \[\lim_{x \rightarrow \infty}\left[ \log_{5}(\frac{ 1 }{ 125 }-2^{-x}) \right] = \log_5\left[ \color{blue}{\lim_{x \rightarrow \infty}(\frac{ 1 }{ 125 }-2^{-x})} \right]\]

  3. anonymous
    • one year ago
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    oh, okay

  4. anonymous
    • one year ago
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    What do I do next?

  5. rational
    • one year ago
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    you can do many things maybe just think of what happens to \(\large 2^{-x}\) as you make \(x\) large

  6. rational
    • one year ago
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    \(2^{-1} = ?\) \(2^{-2} = ?\) \(2^{-3} = ?\) \(\cdots\) \(2^{-100} = ?\)

  7. anonymous
    • one year ago
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    Doesn't the value become smaller?

  8. rational
    • one year ago
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    evaluate those values and see

  9. anonymous
    • one year ago
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    .5 .25 .125 7.8888...E-31

  10. rational
    • one year ago
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    you can see the value of \(2^{-x}\) is approaching \(0\) as you increase \(x\) so \[\lim\limits_{x\to\infty}2^{-x} = 0\]

  11. rational
    • one year ago
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    \[\begin{align} \lim_{x \rightarrow \infty}\left[ \log_{5}(\frac{ 1 }{ 125 }-2^{-x}) \right] &= \log_5\left[ \color{blue}{\lim_{x \rightarrow \infty}(\frac{ 1 }{ 125 }-2^{-x})} \right]\\~\\ &=\log_5\left[ \color{blue}{\frac{ 1 }{ 125 }-0} \right]\\~\\ &=\log_5\left[ \color{blue}{5^{-3}} \right]\\~\\ &=-3 \end{align}\]

  12. anonymous
    • one year ago
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    oohh! I see. Thanks for explaining this. Greatly appreciated! :) I'm not really good at limits and thinking about infinities and such. Any tips?

  13. rational
    • one year ago
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    my only tip is not to try and visualize everything, sometimes you need to just follow the rules and things will be simple

  14. rational
    • one year ago
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    not meant to say, stop visualizing... just want to say that following rules is also important as calculus is very huge, learning wont be smooth w/o a systematic approach graph everything but don't always try to understand in terms of graphs only https://www.desmos.com/calculator

  15. anonymous
    • one year ago
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    Oh, okay. I will keep that in mind. Thanks for everyting! :)

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