LIMITS

LIMITS

                              A  limits  is  a  value  that  a  function  or  sequences  "approaches" as  the  input  or  index  approaches  same  value  .  Limits  are  essential  to  calculus  and  are  used  to  defined  continuity, derivatives  and  integrals.

           

LIMITS  OF  A  FUNCTION

      Suppose  f  is  a  real  valued  function  and  c  is  a  real  number.  then  the  expression,

             It  means  that  f(x) can  made  to  be  as  close  to  L  by  desired  making  X  sufficiently  close  to  c.  it  means  that  above  equation  can  be  made  as  "the  limits  of  F  of  X,as  approaches  to  c,that  is  L.

LIMITS  OF  A  SEQUENCE  

Considered  the  following  sequences,  1.79,  1.799,  1.7999.........  it  can  be  observed  that  number  are  approaches  to  1.8 ,  the  limit  of  the  sequences.Formally  suppose  that

         a1, a2,  are  the  sequences  of  a  real  number.It  can  be  stated  that  a  real  number is  L of the  limit  of  this  sequence.

       

Which  is  read  as

    "The  limit  of  an  as  n  approaches  infinity  equals L"

INDEFINITE INTEGRAL , DIFFERENTIATION , DEFINITE INTEGRAL

INDEFINITE INTEGRAL 

FORMULAS

DIFFERENTIATION   FORMULAS 

DEFINITE  INTEGRAL  FORMULAS

Trigonometry Formulas

• sin²x + cos²x = 1
• tan²x + sec²x = 1
• cosec²x + cot²x = 1
• sin2x = 2sinxcosx
• sinx = 2sinx|2cosx|2
• cos2x = 2cos²x-1
• cos2x = 1-2sin²x
• cos2x = 1 - tan²x/1 + tan²x
• cos2x = cos²x - sin²x
• sin2x = 2tanx/1 + tan²x
• 1 - cosx = 2cos²x/2
• 1 + cosx = 2sin²x/2
• tan2x = 2tanx/1-tan²x
• sin3x = 3sinx - 4sin³x
• cos3x = 4cos³x - 3cosx
• tan3x = 3tanx - tan³x/1 - 3tan²x
• sin(x+y) = sinxcosy + cosxsiny
• sin(x-y) = sinxcosy - cosxsiny
• cos(x+y) = cosxcosy - sinxsiny
• cos(x-y) = cosxcosy + sinxsiny
• tan(x+y) = tanx + tany/1 - tanxtany
• tan(x-y) = tanx - tany/1 + tanxtany
• cot(x+y) = cotxcoty - 1/cotx + coty
• cot(x-y) = cotxcoty + 1/coty - cotx
• sin(x+y) * sin(x-y) = sin2x - sin2y or cos2y - cos2x
• cos(x+y) * cos(x-y) = cos2x - sin2y or cos2y - sin2x
• sin(x+y) + sin(x-y) = 2sinxcosy
• sin(x+y) - sin(x-y) = 2cosxsiny
• cos(x+y) + cos(x-y) = 2cosxcosy
• cos(x-y) - cos(x+y) = 2sinxsiny
• sinC + sinD = 2sin(C+D)/2*cos(C-D)/2
• sinC - sinD = 2cos(C+D)/2*sin(C-D)/2
• cosC + cosD = 2cos(C+D)/2*cos(C-D)/2
• cosC - cosD = 2sin(C+D)/2*sin(D-C)/2