Lim gif of tanx by x as x tend to 0

Dear Vikash,

lim x-->0 [tan x /x] where [] is the greatest integer function(gif)
Considering f(x)=tanx - x
f'(x)=secx^2 -1
     =tanx^2
Thus f'(x)>0
Thus it is an increasing function.
f(0)=0
So for x>0 f(x)>0
or tan x > x
or tan x /x >1

Thus value  of [tanx /x] as lim x-->0+ is 1,where [] is the greatest integer function(gif)

[tanx /x] as lim x-->0- is also 1 as -ve sign in both the numerator as well as denominator

gets cancelled out.

Thus Lim x-->0+ [tan x /x]= lim x-->0- [tan x /x]= 1.

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