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How to prove that the limit does not exist for the following function

$$\lim_{x\to +\infty} \frac{\sin x - x}{\sin x + \cos x}$$ I think using the definition or the sequential characterization are a way to prove that.

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    $\begingroup$ The first requirement for the limit of something as $x \to +\infty$ to exist is that it is defined for all sufficiently large $x$. Your function is undefined for $x = (n-1/4) \pi$ where $n$ is an integer, as denominator is $0$. $\endgroup$ – Robert Israel Sep 6 at 22:51
  • $\begingroup$ @RobertIsrael I think you mean for arbitrarily large $x.$ $\endgroup$ – Allawonder Sep 6 at 23:19
  • $\begingroup$ @Allawonder : No, sufficiently large is correct. It implies “arbitrarily large”. $\endgroup$ – MPW Sep 6 at 23:54
  • $\begingroup$ @MPW Oh, I just looked again and realised he said all sufficiently large $x.$ In that case it does indeed imply for arbitrarily large $x.$ $\endgroup$ – Allawonder Sep 7 at 14:39
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Define $f(x)={\sin x-x\over \sin x+\cos x}$. For $x_n=2\pi n$ we have $$f(x_n)=-x_n\to -\infty$$and for $x_n=(2n+1)\pi$ we have $$f(x_n)=x_n\to \infty$$therefore the limit does not exist.

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