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

  • $\begingroup$ Try with a different method: the left hand limit does equal the right hand limit in this case $\endgroup$ – Aniruddha Deb Jan 24 at 16:02
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    $\begingroup$ 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 $[3\sin(x)/x]$? $\endgroup$ – Ian Jan 24 at 16:02
  • $\begingroup$ @Ian Got it It will be some value less than 3 but greater than 2 so the limit will equal 2 Thanks $\endgroup$ – 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 $x$.


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