I've been learning Squeeze Theorem (and limits in general), but am having problems understanding how to apply it. I understand the basics of the theorem (I think), but I've come across a problem that I'm not even sure how to start solving. I realize that Squeeze Theorem is the way to solve it, but beyond that, I'm clueless.
I've search around the site, and a few other places online, but I can't seem to find a similar problem.
So, here's my problem:
$$\lim_{x\to 0} \frac{3 - \sin(e^x)}{\sqrt{x^2 + 2}}$$
Looks easy enough, but I'm clearly missing something obvious. How do we approach a problem like this? I'd love to show my work, but I'm not sure where to start.
After my best attempts, I believe this is the most appropriate answer to my question.
The problem cannot be solved by Squeeze Theorem, because:
$$\frac{2}{\sqrt{x^{2} + 2}} \leq \frac{3 - \sin{(e^{x})}}{\sqrt{x^{2} + 2}} \leq \frac{4}{\sqrt{x^{2} + 2}}$$
and
$$\lim_{x\to0} \frac{2}{\sqrt{x^{2} + 2}} \neq \lim_{x\to0} \frac{4}{\sqrt{x^{2} + 2}}$$