Greatest Limit

Well, there are much better ways to solve this problem (using Squeeze theorem, some inequalities etc.) but I would show a graphical approach.

Well, can you make the graph of $\frac{x}{\sin(x)}$ and $\frac{\sin(x)}{x}$ ? Not directly obviously but one can guess how their graphs would look like. They are just $x \csc(x)$ and $sinc(x)$ and we know the graphs of $\sin(x)$ and $\csc(x)$ and the $x$ s will slightly change the graph in width etc. but the shape will remain quite a bit same.

I shall show you the graphs of both our functions and then I hope you'll realize that it is really guessable(Guesses are really important in science and mathematics, said Mr. Feynman(not exact words though))

$\frac{\sin(x)}{x}$ -

$\frac{x}{\sin(x)}$ -

Now, as you can see that in the first function when it is approaching 0, it is coming from the bottom and reaching the maximum and hence, greatest integer function would give a lesser value.

Similarly, in the second function, the graph is coming down to its local minima while approaching 0 and hence, greatest integer function should give a greater value.

Now, we can move to the problem,

$\displaystyle \lim_{x\rightarrow 0} \left( \left \lfloor \frac{100 x}{\sin x}\right \rfloor + \left \lfloor \frac{99 \sin x}{x}\right \rfloor \right) = 100 \lim_{x \rightarrow 0}\left(\frac{x}{\sin(x)}\right) + (99-1)\lim_{x \rightarrow 0}\left(\frac{\sin(x)}{x}\right)$ [ $1$ is subtracted since as we discussed above, the greatest integer must be less and the greatest integer less than $99$ is $98$ ]

hence, $\displaystyle = 100 + 98 = \boxed{198}$

I present here an informal solution that allows you to quickly arrive at the correct conclusion.

For small x, $\sin(x) \approx x$ . To evaluate the floor functions, we need to know whether $\sin(x)$ is slightly greater than or less than x. If $sin(x) < x$ then the sum is 100 + 98. If $sin(x) > x$ then the sum is 99 + 99. For x slightly less than 0, one of these will be true, and for x just greater than 0, the other will be true. In either case the sum is 198.

So, the limit exists and equals 198.