Property 18
Limit of Greatest Integer Function
\[f(x)=\lfloor g(x)\rfloor,\qquad g(a)\in\mathbb Z\]
When the inner expression reaches an integer, that point is critical and one-sided limits are analyzed. Away from such critical points the greatest-integer function is locally constant.
1.
\(\lim_{x\to3/2}\lfloor x\rfloor=?\)
2.
\(\lim_{x\to2}\lfloor x\rfloor=?\)
3.
\(\lim_{x\to-4^-}\lfloor x\rfloor=?\)
4.
\(\lim_{x\to-3}\lfloor\samefrac x2\rfloor=?\)
5.
\(\lim_{x\to-1/2}\lfloor x\rfloor=?\)
6.
\(\lim_{x\to\pi/2^+}(\operatorname{sgn}(\tan x)+\lfloor|\cos x|\rfloor)=?\)
7.
\(\lim_{x\to4}\sqrt{|x+2|+\operatorname{sgn}(x+2)}=?\)
8.
\(\lim_{x\to2^+}|2x-\lfloor x+\samefrac12\rfloor|=?\)
9.
\(\lim_{x\to2^-}\lfloor x-2\rfloor=?\)
10.
\(\lim_{x\to3/2}(\lfloor x\rfloor-\operatorname{sgn}(-x))=?\)
11.
\(\lim_{x\to2^+}\left\lfloor\samefrac{|9x-2|}{2}\right\rfloor=?\)
12.
\(\lim_{x\to5^+}\left\lfloor\samefrac{1-3x}{2}\right\rfloor=?\)
13.
\(\lim_{x\to-\infty}\samefrac{\lfloor x\rfloor}{x}=?\)