# Tagged Questions

Used for questions and equations involving the floor function, which is defined to be the function that returns the largest integer less than or equal to $x$ (often denoted by $\lfloor x\rfloor$). See also (ceiling-function).

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### Evaluate $\int_{0}^{\infty} (-1)^{\lfloor x\rfloor}\cdot e^{-x} dx$ [closed]

I'm having trouble integrating the following: $$\int_{0}^{\infty} (-1)^{\lfloor x \rfloor}\cdot e^{-x} \, \mathrm{d}x$$ where $\lfloor x \rfloor$ denotes the floor of $x$. Can you help please?
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### How can I remove the floor function from ⌊ab/10⌋?

After several weeks of trying on my own, I was hoping for a hand. I am familiar with transitioning ⌊ab/10⌋ to a mod function as well as a trig function. Ideally, I would like a solution that involved ...
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### Removing dicontinuity from functions involving modulo?

I am currently looking into removing discontinuity from piecewise continuous functions without changing the derivative where it is defined and (preferably) the value of right sided limit at 0. This is ...
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### Removing jump discontinuity from a tricky function.

I have the function $\cos(x)\lfloor x \rfloor$ which I would like to make continuous without changing the derivative where it exists or the values approaching 0 from the right side. I can do this by ...
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### Example of a jump discontinuity where the left and right hand limits do not exist? [closed]

Right off the bat I should probably mention that I am speaking more visually rather than in manners that can be proven rigorously. Please keep that in mind when reading. I'm looking for a function ...
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### Is the limit $\lim_{x\rightarrow0}\frac{\sin{[x]}}{[x]}$ a one sided limit or not?

Is the limit $\displaystyle\lim_{x\rightarrow0}\frac{\sin{[x]}}{[x]}$ a one sided limit or not? Here $[\, \cdot\, ]$ is the greatest integer function. According to me the right hand limit will be not ...
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### Property of the floor function

Given $u,v \in \mathbb{R}_{+}$, and let $n:=\lfloor v \rfloor$. where $u\in [0,1]$. Is $\lfloor vu \rfloor =\lfloor nu \rfloor$ or $\lfloor vu \rfloor =\lfloor nu \rfloor+1$? Edit: I missed the ...
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