Question

Prove that `[lim_(xto0) (tan^(-1)x)/(x)]=0,` where `[.]` represents the greatest integer function.

Answer 1

See the graph of y=x and `tan^(-1)x` in the figure:

From the figure, when `xto0^(+)`, graph of y=x is above the graph
So,`" "tan^(-1)xltx" or "(tan^(-1)x)/(x)lt1`
`implies" "underset(xto0^(+))lim(tan^(-1)x)/(x)=1^(-)`
`implies" "[underset(xto0^(+))lim(tanx)/(x)]=0`
Thus`" "[underset(xto0)lim(tan^(-1)x)/(x)]=0`