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

Explanation

We know that greatest integer function changes its value at integral values of x.
So, let us find LHL and RHL.
`LHL=underset(xto2)lim[x]=underset(htoo)lim[2-h]`
Sine h is positive and infinitely small `(2-h)` is value very close to 2 but smaller than 2 (say 1.99999)
`:." "[2-h]=1,` when h approaches to zero. Thus, LHL=1
`RHL=underset(xto2)lim[x]=underset(hto0)lim[2+h]=[2.00001]=2`
We observe that `LHLneRHL.` So, `underset(xto2)lim[x]` does not exist.