Given a real valued function $f$ such that $f\left(x\right)=\left\{\begin{array}{l} \\ \\ \\ \end{array}\right.\begin{array}{ll}\frac{{\tan }^2\left\{x\right\}}{(x^2-[x{]}^2)}& \text{for }x>0\\ 1& \text{for }x=0\\ \sqrt{\left\{x\right\}\cot \left\{x\right\}}& \text{for }x<0\end{array},$ where $\left[x\right]$ is the integral part and $\left\{x\right\}$ is the fractional part of $x$, then

$\text{We have}\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0^{+}}{lim}\frac{{\tan }^2\left\{x\right\}}{(x^2-[x{]}^2)}$

$=\underset{x\to 0^{+}}{lim}\frac{{\tan }^{2}x}{x^2}=1.....\left(1\right)$

$[\because x\to 0^{+};[x]=0 and \{x\}=x]$

$\text{Also}\underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0^{-}}{lim}\sqrt{\left\{x\right\}\cot \left\{x\right\}}=\sqrt{\cot 1}.....\left(2\right)$

$[\because x\to 0^{-};[x]=-1\Rightarrow \{x\}=x+1\Rightarrow \{x\}\to 1]$

$\text{Also, }{\cot }^{-1}{\left(\underset{x\to 0}{lim}f\right(x\left)\right)}^2={\cot }^{-1}(\cot 1)=1$