Expert: Paul Klarreich - 1/4/2011

Question
find  lim(x→π/2)〖tan2x/(x-π/2)〗

Answer
find lim(x→pi/2)〖tan2x/(x-pi/2)〗

You have:

lim(x→pi/2) of

       sin 2x
----------------------
cos 2x (x - pi/2)

which is:
lim(x→pi/2) of

  1       sin 2x
--------  -------------
cos pi   (x - pi/2)


But cos pi = -1, so :

lim(x→pi/2) of

     sin 2x
-  -------------
   (x - pi/2)

which is: lim(x→pi/2) of

   2 sin x cos x
-  --------------
   (x - pi/2)

But sin pi/2 = 1:

which is: lim(x→pi/2) of

   2 cos x
-  ------------
   (x - pi/2)

which is: lim(x→pi/2) of

   2 cos x
+ -----------
  (pi/2 - x)

which is: lim(x→pi/2) of

 2 sin (pi/2 - x) << complementary angles rule
+ ----------------
   pi/2 - x

Which is, assuming we set y = pi/2 - x,

lim(y→0) of

 2 sin y
+ ---------
   y

which is 2(1) = 2