Unnamed Article 33

sin Image analysis of oscillation function

about \(\sin\frac{1}{x}\), When \(x\) from 0 Close to the right side of 0 when ,\(\frac{1}{x}\) The change is very big , For example, when \(x\) from \(\frac{1}{10}\) Smaller to \(\frac{1}{100}\), just \(\frac{9}{100} = \mathrm{\Delta}x = \frac{1}{10}\)-\(\ \frac{1}{100}\) Changes will make \(\frac{1}{x}\) from 10 Change to 100, And by the \(\sin 10 = \sin\frac{1}{\frac{1}{10}}\) Change to \(\sin 100 = \sin\frac{1}{\frac{1}{100}}\) In fact, it has crossed several \(sinx\) The cycle of （ Period is \(2\pi \approx 6.18\)）, therefore \(\sin\frac{1}{x}\) stay \(\frac{1}{10}\) and \(\frac{1}{100}\) Many times between -1 and 1 Between . All in all , stay 0 Small near \(x\) Change will cause great \(\frac{1}{x}\) The change of , That would cause \(\sin\frac{1}{x}\) The changes span more \(sinx\) The cycle of , That is to say, this kind of 0 Tiny nearby \(x\) Change will make \(\sin\frac{1}{x}\) stay -1 and 1 Between many times , When \(x\) Getting smaller and smaller ,\(\frac{1}{x}\) The range of change will be larger and larger , So much so that \(\sin\frac{1}{x}\) Images of will be more frequently in -1 and 1 Back and forth or concussion , This is its image in 0 The reason why the neighborhood becomes dense .

When \(x\) When it is positive and getting bigger ,\(\frac{1}{x}\) It's getting smaller , When \(\frac{1}{x}\) Less than \(\pi\) when （ At this time due to \(x\) Positive so \(\frac{1}{x} > 0\)）, Now \(\sin\frac{1}{x}\) The value range of is \(sinx\) On \((0,\pi)\) On the situation —— The function image is positive , In other words, when \(x\) Greater than \(\frac{1}{\pi}\) when \(\sin\frac{1}{x}\) The image of is always positive , As shown in the figure below .

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On the other hand , according to \(\lim_{x \rightarrow 0}\frac{sinx}{x} = 1\) Yes \(\lim_{x \rightarrow \infty}\frac{\sin\frac{1}{x}}{\frac{1}{x}} = 1\), That is to say, when \(x\) When I was very old \(\sin\frac{1}{x}\) and \(\frac{1}{x}\) The value of will be very close , That's why \(\sin\frac{1}{x}\) and \(\frac{1}{x}\) My image is in \(x\) The reason why it will get closer and closer when it gets bigger .

Similarly, we can also explain why \(x\) Very big time \({x^{2}\sin}\frac{1}{x}\) and \(x\) The image of is very close （ In this case \({x^{2}\sin}\frac{1}{x} \approx {x^{2} \times}\frac{1}{x} = x\)）.

about \(x\) It's negative , We can simply according to \(\sin\frac{1}{x}\) Get the corresponding case for the property of odd function .

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1. Introduction to Calculus and Analysis Volume I, Reprint of the 1989 edition, Richard Courant, Fritz John, p38︎