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Evaluate : [underset(x to 0)lim (sin x)/...

Evaluate : `[underset(x to 0)lim (sin x)/(x)]`, where `[*]` represents the greatest integer function.

Text Solution

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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`
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