# limit of floor function

I can solve the question limit of function like $$\lim\limits_{x\to\infty}\frac{\lfloor x-3\rfloor}{x-1}$$ but I cant solve the question like $$\lim\limits_{x\to n^\pm}\frac{\lfloor x-1\rfloor}{x-1}\\ \lim\limits_{x\to n^\pm}\frac{\lfloor x\rfloor}{x-1}$$ Please help me.

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I am assuming that these are the questions your are asking and that $n$ is an integer.

As $x$ approaches $n$ from below, $\lfloor x-1\rfloor=n-2$; therefore, $$\lim_{\large x\to n^-}\frac{\lfloor x-1\rfloor}{x-1}=\frac{n-2}{n-1}$$ As $x$ approaches $n$ from above, $\lfloor x-1\rfloor=n-1$; therefore, $$\lim_{\large x\to n^+}\frac{\lfloor x-1\rfloor}{x-1}=\frac{n-1}{n-1}$$ With these as examples, try the others.

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