Expert: Paul Klarreich - 9/1/2010

Question
lim
x-0   tanx-sinx/x*x*x

Answer
Questioner:   suman dhakal
Category:  Calculus
Private:  no
 
Subject:  limits
Question:  lim x-0 tanx-sinx/x*x*x

tan x-sin x
-------------
  x*x*x

sin x(1/ cos x - 1)
--------------------
  x*x*x
  
sin x(1- cos x)
-----------------
  x*x*x  cos x

sin x(1- cos x)(1 + cos x)
-----------------------------
  x*x*x  cos x(1 + cos x)

 sin x(1- cos^2 x)
----------------------------
 x*x*x  cos x(1 + cos x)

 sin x(sin^2 x)
----------------------------
 x*x*x  cos x(1 + cos x)

 sin^3 x
----------------------------
 x*x*x  cos x(1 + cos x)

 sin x    sin x     sin x                  1
-------  -------  -------  --------------------
   x         x          x        cos x(1 + cos x)

Now lim(x -> 0):
                                         1
  1    *    1   *     1  *   -----------  
                                  1(1 + 1)

= 1/2