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anonymous

  • one year ago

Find the limit: lim as x approaches 0 of cos(1/x)

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  1. anonymous
    • one year ago
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    DNE

  2. SolomonZelman
    • one year ago
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    \(\large\color{slate}{\displaystyle\lim_{x \rightarrow ~0}\cos\left(\frac{1}{x}\right)}\)

  3. SolomonZelman
    • one year ago
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    That, does not exist

  4. anonymous
    • one year ago
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    @pgpilot326 can you explain?

  5. anonymous
    • one year ago
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    @solomonzelman why?

  6. Zale101
    • one year ago
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    If you sub in x=0, you'll get a 0 in the denominator which is undefined.

  7. SolomonZelman
    • one year ago
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    \(\large\color{slate}{\displaystyle\lim_{x \rightarrow ~0}\cos\left(\frac{1}{x}\right)}=\cos\left(\displaystyle \lim_{x \rightarrow ~0}\frac{1}{x}\right)\)

  8. anonymous
    • one year ago
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    as x approaches 0, 1/x approaches infinity. cos will cycles through all of it's values and not settle on a single value (which it would need to do in order for the limit to exist)

  9. SolomonZelman
    • one year ago
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    yes, that is equivalent of \(\large\color{slate}{\displaystyle\lim_{x \rightarrow ~0}\cos\left(\frac{1}{x}\right)}=\cos\left(\displaystyle \lim_{x \rightarrow ~0}\frac{1}{x}\right)=\cos\left(\displaystyle \lim_{x \rightarrow ~\infty }x\right)\)

  10. SolomonZelman
    • one year ago
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    So it will alternate between 1 and -1

  11. SolomonZelman
    • one year ago
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    |dw:1441658924817:dw|

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