Why this limit does not exist?

[Vali]

Why limit e^(1/x) * sinx as x->0 does not exist?
My approach: -1<= sinx <= 1 / *e^(1/x) so my limit would would have two limits (-+infinity) so the limit doesn't exist.Is it correct?

 

[pka]

Why limit e^(1/x) * sinx as x->0 does not exist?
My approach: -1<= sinx <= 1 / *e^(1/x) so my limit would would have two limits (-+infinity) so the limit doesn't exist.Is it correct?

Well \(\displaystyle \mathop {\lim }\limits_{x \to {0^ - }} \exp ({x^{ - 1}})\sin (x) = 0\) & \(\displaystyle \mathop {\lim }\limits_{x \to {0^ + }} \exp ({x^{ - 1}})\sin (x) = \infty\). Thus the limit does not exist. SEE HERE

 

[Vali]

Yes, you're right.
Thank you for your help.