TITLE: What type of discontinuity is $\sin(1/x)$?
QUESTION [8 upvotes]: For those of you familiar with the graph of $\sin(\dfrac{1}{x})$, things get quite 'intense' as $x \to 0$.
Is this a removable discontinuity or an infinite discontinuity or even discontinuous at all?
Lastly, is it differentiable at $x = 0$?

REPLY [4 votes]: The function $\sin{\frac{1}{x}}$ Does not attain a limit as $x\rightarrow 0$. 
Let $f(x)=\sin{\frac{1}{x}}$
$f(x)$ attains a limit $l$ then there exists a $\delta>0$ for every $\epsilon >0$ such that 
$\left|f(x)-l\right|<\epsilon$ such that $0<\left|x-0\right|<\delta$.
Considering it from the right we have $0<x<\delta$ or $\infty<\frac{1}{x}<\frac{1}{\delta}$
. Which means There are infinite number of $\frac{1}{x}$ of the form $2n\pi$. Which means the function is oscillating in the interval and  limit does not  exist