Question

Find $\underset{x\to 3^{+}}{\lim }\frac{x}{\left[x\right]}.$ Is it equal to $\underset{x\to 3^{-}}{\lim }\frac{x}{\left[x\right]}.$

Answer 1

$$\begin{gathered} \underset{x\to 3^{+}}{\lim }\frac{x}{\left[x\right]} \\ \mathrm{Let}x=3+h,\mathrm{where}h\to 0. \\ \Rightarrow \underset{h\to 0}{\lim }\left(\frac{3+h}{\left[3+h\right]}\right)=\frac{3}{3}=1 \\ \mathrm{Also},\underset{x\to 3^{-}}{\lim }\left(\frac{x}{\left[x\right]}\right) \\ \mathrm{Let}x=3-h,\mathrm{where}h\to 0. \\ \underset{h\to 0}{\lim }\left(\frac{3-h}{\left[3-h\right]}\right) \\ =\frac{3}{2} \\ \therefore \underset{x\to 3^{-}}{\lim }\frac{x}{\left[x\right]}\ne \underset{x\to 3^{+}}{\lim }\frac{x}{\left[x\right]} \end{gathered}$$