anonymous
  • anonymous
Evaluate. \[\lim_{x \rightarrow \infty}\left[ \log_{5}(\frac{ 1 }{ 125 }-2^{-x}) \right]\]
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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rational
  • rational
log is continuous function, so you can send the limit inside the function : \[\lim(\log(f(x))) = \log(\lim f(x))\]
rational
  • rational
\[\lim_{x \rightarrow \infty}\left[ \log_{5}(\frac{ 1 }{ 125 }-2^{-x}) \right] = \log_5\left[ \color{blue}{\lim_{x \rightarrow \infty}(\frac{ 1 }{ 125 }-2^{-x})} \right]\]
anonymous
  • anonymous
oh, okay

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anonymous
  • anonymous
What do I do next?
rational
  • rational
you can do many things maybe just think of what happens to \(\large 2^{-x}\) as you make \(x\) large
rational
  • rational
\(2^{-1} = ?\) \(2^{-2} = ?\) \(2^{-3} = ?\) \(\cdots\) \(2^{-100} = ?\)
anonymous
  • anonymous
Doesn't the value become smaller?
rational
  • rational
evaluate those values and see
anonymous
  • anonymous
.5 .25 .125 7.8888...E-31
rational
  • rational
you can see the value of \(2^{-x}\) is approaching \(0\) as you increase \(x\) so \[\lim\limits_{x\to\infty}2^{-x} = 0\]
rational
  • rational
\[\begin{align} \lim_{x \rightarrow \infty}\left[ \log_{5}(\frac{ 1 }{ 125 }-2^{-x}) \right] &= \log_5\left[ \color{blue}{\lim_{x \rightarrow \infty}(\frac{ 1 }{ 125 }-2^{-x})} \right]\\~\\ &=\log_5\left[ \color{blue}{\frac{ 1 }{ 125 }-0} \right]\\~\\ &=\log_5\left[ \color{blue}{5^{-3}} \right]\\~\\ &=-3 \end{align}\]
anonymous
  • anonymous
oohh! I see. Thanks for explaining this. Greatly appreciated! :) I'm not really good at limits and thinking about infinities and such. Any tips?
rational
  • rational
my only tip is not to try and visualize everything, sometimes you need to just follow the rules and things will be simple
rational
  • rational
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
anonymous
  • anonymous
Oh, okay. I will keep that in mind. Thanks for everyting! :)

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