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

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rational
 one year ago
Best ResponseYou've already chosen the best response.3log is continuous function, so you can send the limit inside the function : \[\lim(\log(f(x))) = \log(\lim f(x))\]

rational
 one year ago
Best ResponseYou've already chosen the best response.3\[\lim_{x \rightarrow \infty}\left[ \log_{5}(\frac{ 1 }{ 125 }2^{x}) \right] = \log_5\left[ \color{blue}{\lim_{x \rightarrow \infty}(\frac{ 1 }{ 125 }2^{x})} \right]\]

rational
 one year ago
Best ResponseYou've already chosen the best response.3you can do many things maybe just think of what happens to \(\large 2^{x}\) as you make \(x\) large

rational
 one year ago
Best ResponseYou've already chosen the best response.3\(2^{1} = ?\) \(2^{2} = ?\) \(2^{3} = ?\) \(\cdots\) \(2^{100} = ?\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Doesn't the value become smaller?

rational
 one year ago
Best ResponseYou've already chosen the best response.3evaluate those values and see

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0.5 .25 .125 7.8888...E31

rational
 one year ago
Best ResponseYou've already chosen the best response.3you can see the value of \(2^{x}\) is approaching \(0\) as you increase \(x\) so \[\lim\limits_{x\to\infty}2^{x} = 0\]

rational
 one year ago
Best ResponseYou've already chosen the best response.3\[\begin{align} \lim_{x \rightarrow \infty}\left[ \log_{5}(\frac{ 1 }{ 125 }2^{x}) \right] &= \log_5\left[ \color{blue}{\lim_{x \rightarrow \infty}(\frac{ 1 }{ 125 }2^{x})} \right]\\~\\ &=\log_5\left[ \color{blue}{\frac{ 1 }{ 125 }0} \right]\\~\\ &=\log_5\left[ \color{blue}{5^{3}} \right]\\~\\ &=3 \end{align}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0oohh! I see. Thanks for explaining this. Greatly appreciated! :) I'm not really good at limits and thinking about infinities and such. Any tips?

rational
 one year ago
Best ResponseYou've already chosen the best response.3my only tip is not to try and visualize everything, sometimes you need to just follow the rules and things will be simple

rational
 one year ago
Best ResponseYou've already chosen the best response.3not 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
 one year ago
Best ResponseYou've already chosen the best response.0Oh, okay. I will keep that in mind. Thanks for everyting! :)
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