TITLE: The behavior of the graph of $f(x) = \sin (\pi/x)$ as x approaches 0, Why?
QUESTION [5 upvotes]: This is my first post.  I hope it is relevant.
for the $\lim_{x\to0}\sin(\pi/x)$ The limit does not exist.  
I am curious if my logic is appropriate or if there is another way to understand this. 
So what I believe is the following:   
As $x\to 0$ we have that $  \pi/x\to\infty$  
Therefore,  $\sin(\infty)$, which makes sense by the fact that as x approaches 0, the input of sine will increase to infinity or some large number.  As a result, sine will repeat its periods indefinitely.  It will oscillate between 1 and -1 indefinitely as x approaches 0.  
I missed one thing and that is this is a two-sided limit, so sine's input is approaching positive and negative infinity

REPLY [2 votes]: Graph of sin(π/x) with x axis changing from +/- 1 units to +/- 0.01 units. This shows graphically the intuition the OP had that the frequency of the sine wave approaches infinity.