Question

Prove that $\underset{x\to a^{+}}{\lim }\left[x\right]=\left[a\right]$ for all a ∈ R. Also, prove that $\underset{x\to 1^{-}}{\lim }\left[x\right]=0.$

Answer 1

$$\begin{gathered} \underset{x\to a^{+}}{\lim }\left[x\right] \\ \mathrm{Let}x=a+h,\mathrm{where}h\to 0. \\ \underset{h\to 0}{\lim }\left[a+h\right] \\ =a \\ \underset{x\to 1^{-}}{\lim }\left[x\right] \\ \mathrm{Let}x=1-h,\mathrm{where}h\to 0. \\ \underset{h\to 0}{\lim }\left[1-h\right] \\ =0 \end{gathered}$$