In an attempt to solve this question, I managed to rewrite the given limit as
$$\frac{1}{\sqrt{2}}\lim_{x \to \infty} \left(\pi \lfloor x \rfloor^{3/2} - 2\sqrt{\lfloor x \rfloor} - \frac{4}{\sqrt{\lfloor x \rfloor}}\sum_{k = 1}^{\lfloor x \rfloor - 1} \sqrt{\lfloor x \rfloor^2 - k^2}\right)$$
To evaluate the limit, I essentially need to find the horizontal asymptote of
$$f(x) = \pi \lfloor x \rfloor^{3/2} - 2\sqrt{\lfloor x \rfloor} - \frac{4}{\sqrt{\lfloor x \rfloor}}\sum_{k = 1}^{\lfloor x \rfloor - 1} \sqrt{\lfloor x \rfloor^2 - k^2}$$
While I would obviously love to know how to compute something like this, I am seeking a more general strategy to find the horizontal asymptote of a function involving the floor function.
The following are some of my ideas:
- My first idea is to create a set of data points, i.e. $(x, f(x))$, and then use Newton interpolation to model a polynomial that passes through the points. Unfortunately, this method is limited by the fact that I can only collect a finite number of points and it also won't be asymptotic in nature, perhaps crossing the asymptote at many instances. Maybe there is a way to smoothen the curve below the asymptote and compute some kind of maxima, but this approach doesn’t seem very promising to me.
- My second idea is to play around with other similarly asymptotic continuous functions until I eventually stumble upon one that has the same asymptote as the original function. The first problem with this approach is that it is like a shot in the dark but perhaps this can be eliminated with some educated guesses. The second problem is that even if I stumble upon a function with the same horizontal asymptote, I won't have a way of knowing that the asymptotes for the two functions match, except for probably some weak graphical intuition. For example, I highlight $3$ functions that have the same horizontal asymptote but different rates of increase below:
- My third idea is to view $f(x) < a$ as an inequality with $a$ being the desired horizontal asymptote. Since the floor function is closely related to inequalities, this may be a viable approach. The problem, I believe, is that I can't be sure that $a$ is the horizontal asymptote of the function but I can perhaps keep getting closer to the smallest value of $a$ that satisfies the inequality.
- My fourth idea is to view the floor function as points with horizontal error bars. One can perhaps construct a computational algorithm that finds the best fit line that passes through all error bars. My biggest problem with this method is that it takes the aid of software and I would like to restrict the method to be as mathematical as possible. Nonetheless, I think another problem is that I can only use a finite set of points and in the best case, I will only have a good numerical approximation.
- My fifth idea is to model the distances between $2$ discrete horizontal lines of the function and try to find some converging sum of the distances. This can then be added to the $y$-coordinate of the lowest horizontal line of the function to compute the horizontal asymptote.
Perhaps there isn't a good way to go about doing this other than numerical approximations. But my knowledge of limits and floor functions is too limited for me to make any conclusions. I appreciate any and all insights.