Prove that `[lim_(xto0) (sinx)/(x)]=0,` where `[.]` represents the greatest integer function.

Explanation

See the graphs of y=x and `sin x` in the following figure.

From the figure when `xto0^(+)`,graph of y=x is above the graph of `y=sinx`
i.e.,`" "sinxltx" or " (sinx)/(x)lt1`
`implies" "underset(xto0^(+))lim(sinx)/(x)=1^(1-)`
`implies" "[underset(xto0^(+))lim(sinx)/(x)]=0`
When `xto0^(-)`, graph of y=x is below the graph of `y=sinx`
i.e., `sinxltx" or " (sinx)/(x)lt1" "`(as x is negative)
`implies" "underset(xto0^(-))lim(sinx)/(x)=1^(-)`
`implies" "[underset(xto0^(-))lim(sinx)/(x)]=0`
Thus, `[underset(xto0^(-))lim(sinx)/(x)]=0`