Getting the limits, we have the following:
[tex]\begin{gathered} \lim_{x\rightarrow0^+}\Big(\frac{|x|}{x}\Big)=1,\text{ \lparen however two sided limit does not exist\rparen} \\ \\ \lim_{x\rightarrow0}\sin\Big(\frac{1}{x^2}\Big)=DNE \\ \\ \lim_{x\rightarrow0}\frac{x^3}{|x|}=0 \\ \\ \lim_{x\rightarrow0}\frac{1}{x-1}=-1 \end{gathered}[/tex]
Therefore, the limit does not exist for option 2.
