Will the limit exist? If $$f(x) = \begin{cases} \frac{\sin([x]+x)}{[x]+x} &\textrm{if } x \neq 0, \\ 1 &\textrm{if } x = 0, \end{cases}$$
where [.] is the greatest integer function, then does
$\lim_{x \to 0}f(x)$ exist or not?

I thought of using the property that  $\lim_{x \to 0} \frac{\sin g(x)}{g(x)}=1$ by which I thought that the limit should exist but my book said otherwise. What did I miss? Is there anything else that should be applied?

The left and right limits at $0$ must be equal to $1$ for continuity. But you can see that as $x\to 0^-$, $$\frac{\sin([x] +x)}{[x] + x}= \frac{\sin(-1+x)}{-1+x} \to \sin 1 \ne 1 $$
