Determine the limit of the function $f$ at $0$, for the function defined on $\mathbb{R}$ by $f(0)=0$ and $f(x)=\sin(1/x)$ for $x \neq 0$.

Question

I have a problem of analysis, for which I'd like to ask some tips how to begin, precisely I need to determine the limit of the function, if it's exists, by using a formal definition of a limit (epsilon delta).

The function is defined by the following:

$f(0) = 0 $ if $x = 0$

$f(x) = \sin(1/x)$ if $x \neq 0 $

Domain of the definition of the function is $\mathbb{R}$

Answer 1

We have that the sequence $$x_n=\frac{1}{\left(2n+\frac 12\right)\pi}$$ converges to $0$ as $n\to \infty.$ But

$$f(x_n)=\sin \left(2n+\frac 12\right)\pi=1, \forall n\in\mathbb N.$$

Also the sequence $$y_n=\frac{1}{2n\pi}$$ converges to $0$ as $n\to \infty.$ But

$$f(y_n)=\sin 2n\pi=0, \forall n\in\mathbb N.$$

Thus, the limit doesn't exist.