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I tried using $[x][y]=[xy]$ but clearly, that is wrong

I know that $\lim_{x→0}[\sin(x)x]=0,\lim_{x→0}[\sin⁡(x)x]=0$ and $\lim_{x→0}[\tan(x)x]=1$.

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    Does $[\cdot]$ mean something here? – 2017-01-30
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    You could've used the formatting in your last question and just copied,the way you wrote the question it is ambiguous.Here's a little bit on how to format questions here http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – 2017-01-30
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    The question you intend to ask is entirely unclear. How does this question differ from the duplicate question you asked earlier? – 2017-01-30
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    What is it, exactly, that you are asking: Are you asking about $$ \lim_{x→0}\left[\frac{x^2}{\tan(x)\sin(x)}\right]=0$$ or are you asking about $$\lim_{x→0}\left[\frac{x^2}{\tan(x)} \cdot \sin(x)\right]=0\,?$$ – 2017-01-30

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One can see that

$$\frac{x^2}{\tan(x)\sin(x)}=\frac{\cos(x)}{\left(\frac{\sin(x)}x\right)^2}$$

thus, the limit is $1$ without the floor function $[\cdot]$. With the floor function, we can deduce the limit will be $1$ or $0$, depending on which side $\frac{x^2}{\tan(x)\sin(x)}$ approaches $1$. We can deduce which side the limit falls on by observing the following:

$$\tan(x)\sin(x)-x^2>0\text{ for }|x|0$$

which follows from taking the Maclaurin series expansion.

Thus, the limit is $0$.

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    But $\frac{x^2}{\tan(x) \sin(x)} \to 1$ as $x \to 0$. I think Karmanya meant $\frac{x^2 \sin(x)}{\tan(x)}$ – 2017-01-30
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    @MarvinF Could also be floor function, which is why I asked above in the comments. – 2017-01-30
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    @SimplyBeautifulArt It's the floor function,his last question http://math.stackexchange.com/questions/2121119/does-x-cdot-y-xy-here-is-the-gif-function – 2017-01-30
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    @kingW3 Ah, thanks for the clarification. – 2017-01-30
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    @SimplyBeautifulArt I'm confused. You just wrote a suggested guideline to writing a good answer. This is an ambiguous question, and low quality, at that. Remind me what you suggest regarding answering such questions? – 2017-01-30
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    @amWhy ah, I see your point...the problem is I enjoy answering questions more. – 2017-01-30
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    @Simply Just because you enjoy answering a question such as this, doesn't mean you *should* answer it. If you are sincere about wanting to moderate the site at some point, you should be starting to do so now. You're going to have to put more weight into considering what's best for the site, in the long run, than in finding self-gratification in the moment. The choice is yours, to answer or not. But please do not define a good answer with criteria you don't follow, yourself. "Do as I say, not as I do," doesn't get you far, in terms of gaining the trust of the community. – 2017-01-30
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    @amWhy True enough. :-/ Should I turn it into a community post or something? – 2017-01-30
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    It's hard to know how to best proceed, because the OP has failed to clarify his/her question, in two consecutive posts. I'd just add a word or two about the fact that the question is unclear; but in case you (OP) mean this:............, then .......... Making it a community post would be honorable, but not necessary. If you do wikify it (i.e. make it community wiki), you'll retain the rep you've already garnered up til now. – 2017-01-30
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    @amWhy Hm, that's some weird stuff. I'd expect you to lose the rep, if any, but I guess that doesn't make sense because then people could do some very annoying trolling. – 2017-01-30
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    Sometimes, over the course of time, some posts (entire posts, questions and answers) are "wikified". No one loses any rep they've already earn. But after wikification of an answer or a post in general, no further rep can be acquired. Folks can still upvote such posts (answers or questions), and you'll be informed about upvotes and/or accepts. – 2017-01-30