The lesson I'm doing is solving
$\lim_{x\to0}x^2\sin\left(\dfrac{1}{x}\right)$
with sandwich theorem like so :
$$\displaystyle\lim_{x\to0}-x^2\leq\displaystyle\lim_{x\to0}x^2\sin\left(\dfrac{1}{x}\right)\leq\displaystyle\lim_{x\to 0}x^2$$
When I first look at this though my thought was to use the properties of lim and do this :
$\lim_{x\to0}x^2 = 0$ and times that by $\lim_{x\to0}\sin\left(\dfrac{1}{x}\right)$
And if I'm not an idiot then this says that $0\sin(1/0)$ is going to equal $0$ which is the right answer.
Does that make sense? Instead of bothering with sandwich theorem you could just pull apart the $x^2$ and $\sin(1/x)$, input the $0$ of the limit and not even have to bother with the sine.
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3$\begingroup$ the trouble is $\lim_{x\to 0}\sin(1/x)$ is undefined. $\endgroup$– operatorerrorSep 8, 2019 at 23:31
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$\begingroup$ Right but can't you "skip" that since it will be multiplied by 0? That would seem logical to me. Besides in the lessons they are saying dividing by 0 gives infinity. $\endgroup$– Derek C.Sep 8, 2019 at 23:32
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$\begingroup$ Note that properties of limits don't apply here as one of the factors $x^2$ tends to $0$ and the limit of other factor $\sin(1/x)$ does not exist. So you can either go for sandwich theorem or just use the definition of limit directly. $\endgroup$– Paramanand Singh ♦Sep 9, 2019 at 9:48
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3 Answers
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No, this doesn't work. As
$$\lim_{x\to0}\sin\left(\frac1x\right)$$ does not exist, you may not invoke the product rule.
Using the same reasoning, we could write
$$\lim_{x\to0}\frac xx=\left(\lim_{x\to0}x\right)\left(\lim_{x\to0}\frac 1x\right)\color{red}{=0}$$ "because the first limit is $0$ and we don't care that the second does not exist".
If you claim that "but it will work because the sine is bounded", you are in fact implicitly using a sandwich argument:
$$a<\sin x<b,$$ then
$$ax^2<x^2\sin x<bx^2$$ proves the limit $0$.
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The problem is that $\lim\limits_{x\rightarrow 0}\sin(1/x)=\lim\limits_{X\rightarrow +\infty}\sin(X)$ doesn't exist. You can also write that $|x^2\sin(1/x)|\leqslant x^2\underset{x\rightarrow 0}{\longrightarrow} 0$
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$\begingroup$ But $0 * sin(infinity)$ would equal zero. Doesn't matter if sin(infinity) doesn't really exist, it must be between -1 and 1 right? $\endgroup$– Derek C.Sep 8, 2019 at 23:43
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1$\begingroup$ The fact that $\sin$ is between $-1$ and $1$ is used in the sandwich theorem, what you want to do, the sandwich theorem does it rigorously. You can't manipulate things that doesn't exist... For instance $\lim\limits_{x\rightarrow +\infty}{(\cos x-\cos x)}=0$ but you can't write that $\lim\limits_{x\rightarrow +\infty}{(\cos x-\cos x)}=\lim\limits_{x\rightarrow +\infty}{\cos x}-\lim\limits_{x\rightarrow +\infty}{\cos x}$ because it doesn't make sense, as $1+1=3$ doesn't make sense either. $\endgroup$ Sep 8, 2019 at 23:48
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$\begingroup$ What do you mean? That would make perfect sense. infinity = infinity and cos(x) - cos(x) = 0. And besides 0 x anything is 0. so 0 x sin(any number) = 0 $\endgroup$– Derek C.Sep 8, 2019 at 23:51
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$\begingroup$ Infinity is unpredictable, in fact it is not a number so the statement $0\times\sin(any\ number)=0$ does not apply here. $\endgroup$ Sep 8, 2019 at 23:54
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1$\begingroup$ @DerekC. $0\cdot\text{anything}=0$ is wrong. $\endgroup$– user65203Sep 10, 2019 at 9:25
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I think It make sense. see whatever is in sine function it will always give a value from [-1,1] and if we multiply it by a number which is almost zero we will get zero.
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$\begingroup$ This is a restatement of the sandwich theorem in disguise. $\endgroup$– user65203Sep 10, 2019 at 8:49