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Define Greatest Integer Functions with Examples

GREATEST INTEGER FUNCTIONS GREATER INTEGER FUNCTION of any term represents greatest integer present in that term but not greater than that term. GREATEST INTEGER FUNCTION (but not greater than x) is denoted by y=[x] Graph For Greatest Integer Function Greatest integer function of positive numbers:      1)    Find the greatest integer functions of the following terms                                      i) y=[8.9]  ii) y=[3.4]                 i) y=[8] , greatest integer function is 8                ii)  y=[3.4] , greatest integer function is 3 Note :  from the above cases the greatest integer function of any term represents the integer part of that number Greatest integer function of negative numbers 2)    ...
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LIMIT OF GREATEST INTEGER FUNCTIONS & EXAMPLES

To prove that   lim x⟶0  [sinx/x]=0 , where [.] denotes greatest integer function . We must first prove that sin x/x tends to 1 from the values that are  less than 1 as x tends to zero PROOF : Now as x tends to zero, x can take any of the values tending from right of zero and   tending from left side of zero  Case - 1:                  First we assume that x tends from   right of the zero               Now,  If x= 0.001,   sin(0.001)/0.001     = 0.999999833                         If x=0.0001,  sin(0.0001/0.0001) = 0.999999998   Case -2:                 Now we assume that x is tending from left of the zero                  If   x= -0.001,      sin(-0.00...
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General Solutions of Trigonometric Equations

While determining the number of solutions the equation can have , we use these formulas .   General solutions to some of the standard equations Equation Solution Sin θ Θ  = nπ , n ∈  Z Cos θ Θ  = (2n+1) π /2  ,  n ∈  Z Tan θ  = 0 Θ  = n π  , n ∈  Z Sin θ  = 1 Θ  = (4n+1) π /2  ,  n ∈  Z Sin θ  = -1 Θ  = (4n-1) π /2  ,  n ∈  Z Cos θ  = 1 Θ  = 2nπ , n ∈  Z Cos θ  = -1 Θ  = (2n+1 )π, n ∈  Z Cot θ  = 0 Θ  = (2n+1 ) π /2 ,  n ∈  Z   EQUATIONS OF ANOTHER TYPES           1.    When the equation is of the form sin θ = sin α             General solution to the equation is given by θ = n π + (-1) n , n ∈ Z       2.    When the equation is of the form cos θ =  cos α   ...
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