Expert: Paul Klarreich - 1/26/2011

Question
Why is 6x + 6x cos (6x) / 7sin (6x) cos (6x) as x approaches 0

simplified to

6x / 7sin (6x) cos (6x) + 6x / 7sin(6x) ?

I don't understand the steps involved especially why cos is canceled out in the numerator

Thanks

Answer
Questioner: Jeff
Country: United States
Category: Calculus
Private: No
Subject: limits
Question: Why is 6x + 6x cos (6x) / 7sin (6x) cos (6x) as x approaches 0

simplified to

6x / 7sin (6x) cos (6x) + 6x / 7sin(6x) ?

I don't understand the steps involved especially why cos is canceled out in the numerator

Thanks



6x + 6x cos (6x)
-----------------
7 sin (6x) cos (6x)


Basic limits you will want:

1 - cos x
---------  ---> 0, as x --> 0
  x

sin x
--------  ---> 1, as x --> 0
  x

6x + 6x cos (6x)
------------------- =
7 sin (6x) cos (6x)

6x(1 +  cos (6x))
-------------------
7 sin (6x) cos (6x)

6x(1 +  cos (6x))   (1 -  cos (6x))
------------------- --------------------
7 sin (6x) cos (6x)  (1 -  cos (6x))


6x(1  -  cos^2(6x))
-----------------------------------
7 sin (6x) cos (6x) (1 -  cos (6x))

6x sin^2(6x)
-----------------------------------
7 sin (6x) cos (6x) (1 -  cos (6x))

6x sin(6x)
-----------------------------------
7           cos (6x) (1 -  cos (6x))

 sin(6x)        6x
----------  -----------------
7 cos (6x)   (1 -  cos (6x))


-->

 sin(6x)        6x
----------  -----------------
7 cos (6x)   (1 -  cos (6x))

More to do, of course, but this is a start.