# Limit question about continuity of a given floor function

If $$f(x)= \left[\frac{3 \sin x}{x}\right]$$ where $$\left[ . \right]$$ denotes the greatest integer function. For $$x \neq 0$$ , find the value of $$f(0)$$ so that $$f(x)$$ is continuous at $$x=0$$

I tried solving it using Sandwich theorem but the limits aren't equal

• Try with a different method: the left hand limit does equal the right hand limit in this case – Aniruddha Deb Jan 24 at 16:02
• Consider that for x close to 0, whether on the left or the right, sin(x)/x is close to but slightly less than 1...what does that mean about $[3\sin(x)/x]$? – Ian Jan 24 at 16:02
• @Ian Got it It will be some value less than 3 but greater than 2 so the limit will equal 2 Thanks – user744725 Jan 24 at 16:56

By the Mean Value Theorem, $$\frac{3\sin x-3\sin 0}x=3\cos\xi$$ for some $$\xi$$ between $$0$$ and $$x$$.