# $\lim _{x\to 0}\left(\sin\left(x\right)\cos\left(\frac{5}{x}\right)\right)$ Can I apply Sandwich Theorem to this limit?

Actually the original question was the following

Determine whether the following limit exist?

$$\lim _{x\to 0}\left(x^2\sin\left(\frac{2}{x}\right)+\sin\left(x\right)\cos\left(\frac{5}{x}\right)\right)$$

I know I can apply Sandwich Theorem to the following limit

$$\lim_{x\to 0}\left(x^2\sin\left(\frac{2}{x}\right)\right)$$

but the second part of the function i am not sure, so please help me with this problem. Thank you.

• Note $-|\sin(x)|\le \sin(x) \cos(5/2) \le |\sin(x)|$. – bonsoon Feb 13 at 19:28
• $sin(x)$ is a function that tends to 0 when x>0, and $cos(5/x)$ is bounded between -1 and 1. So It would be a product between an infinitesimal and a bounded function, hence the limit of this product is zero. You can also use sandwich theorem , if you'd like, you could split the limit in a sum of limits. – Jonathan Perales Feb 13 at 19:31
• Okay. That was my opinion too. Yet, some limit calculator sites couldn't find the answer for this question. That's why i asked it to forum in the first place. – HouseBT Feb 13 at 19:32
• You may also try wolframalpha: wolframalpha.com/input/?i=lim+sin(x)+cos(5%2Fx),+as+x+to+0 – bonsoon Feb 13 at 19:33
• @HouseBT Online calculators tend to fail. First you should check if your answer makes sense. If you want to be sure, use a graphing software. It is easy to check the limit of a function that way (most of the time). Don't trust limit calculators, only if they can do the said limit. (They are only useful to check if you are wrong, but if they fail to perform the operation don't be worried). – Jonathan Perales Feb 13 at 19:39