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Does the limit $$\lim_{x \to 0}x^{\cos x}$$ exist? How this limit is different from $$\lim_{x \to 0}|x|^{[\cos x]}$$, where $[x]$ means the greatest integer less than or equal to $x$.

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    $\begingroup$ What does $[\phantom v]$ mean in $[\cos x]$? $\endgroup$ – Arthur Jun 20 at 15:39
  • $\begingroup$ What have you tried? $\endgroup$ – The Count Jun 20 at 15:41
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    $\begingroup$ $[ ]$ means the greatest integer function $\endgroup$ – user677491 Jun 20 at 15:41
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In the first limit $x^{\cos x}$ is not defined if $x$ is negative. In the second limit, $|x|^{\lfloor \cos x \rfloor}$ is always defined. When $|x|$ is very small but not zero, $\lfloor \cos x \rfloor =0$. So now you can figure out both limits.

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