# How to solve this exponential limit?

Does the limit $$\lim_{x \to 0}x^{\cos x}$$ exist? How this limit is different from $$\lim_{x \to 0}|x|^{[\cos x]}$$, where $$[x]$$ means the greatest integer less than or equal to $$x$$.

• What does $[\phantom v]$ mean in $[\cos x]$? – Arthur Jun 20 at 15:39
• What have you tried? – The Count Jun 20 at 15:41
• $[ ]$ means the greatest integer function – user677491 Jun 20 at 15:41

In the first limit $$x^{\cos x}$$ is not defined if $$x$$ is negative. In the second limit, $$|x|^{\lfloor \cos x \rfloor}$$ is always defined. When $$|x|$$ is very small but not zero, $$\lfloor \cos x \rfloor =0$$. So now you can figure out both limits.