Find right and left limit when $x \to \frac{\pi}{3} $

Question

Consider $f(x) = \lfloor \tan^2 x \rfloor $.We want to find $\lim_{x \to (\frac{\pi}{3})^{+}} f(x)$ and $\lim_{x \to (\frac{\pi}{3})^{-}} f(x)$. I know the answers are $3$ and $2$ but I can't find them using inequalities .

Answer 1

NOTE: $\tan ^2 \frac{\pi}{3}=3.$
We know that floor function gives Least integer less than or equal to $x$

Now , $$\lim _{x\rightarrow\frac{\pi^+}{3}} f(x)⌊tan^2x⌋ \text {means giving a value which is less than or equal to our } f(x) \text{ which over here is >3}$$ Thus this value will come out to be 3.

Similarly, when $\lim_{x\rightarrow\frac{\pi^-}{3}}$Then value will be something like $2.999999$ which is equivalent to $2$ (On using floor function) Hope this solves your problem.