Prove $\lim _{x \to 0} \sin(\frac{1}{x}) \ne 0$ Prove $$\lim _{x \to 0} \sin\left(\frac{1}{x}\right) \ne 0.$$
I am unsure of how to prove this problem. I will ask questions if I have doubt on the proof. Thank you!

Here's an intuitive answer, just to shake things up.  The limit of a function as it approaches some point is the value you expect the function to take at that point$^1$ based on what it does nearby the point.  If you can't form any expectations about the function's value, the limit doesn't exist.  
Now take a look at the function in question:

Can anyone possibly form expectations about where that function should be at $0$?
Of course, once you have the intuition, you have to actually write down the proof.  For that, you need to remember the definition of the limit and formalize why you can't figure out where $\sin(x^{-1})$ "should" be at $0$.


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*(Be careful, though!  The function might not even be defined at the point!  We're just forming an expectation of what we think the function should be, if it were to exist.  We don't know anything about what the function actually does.)

