Question

I need help on which one does not exist and why? Please.

Answer 1

Getting the limits, we have the following:

`\begin{gathered} \lim_(x\rightarrow0^+)\Big((|x|)/(x)\Big)=1,\text{ \lparen however two sided limit does not exist\rparen} \\ \\ \lim_(x\rightarrow0)\sin\Big((1)/(x^2)\Big)=DNE \\ \\ \lim_(x\rightarrow0)(x^3)/(|x|)=0 \\ \\ \lim_(x\rightarrow0)(1)/(x-1)=-1 \end{gathered}`

Therefore, the limit does not exist for option 2.