How does (-1/2 * r^3)/(r * sqrt(1 - (1/4 * r^2)) - 1) come out to 2(sqrt(1 - (1/4) * r^2) + 1) ?
Update:
I'm doing a limits problem and trying to get an answer that doesn't resolve to 0/0. The answer I posted is the correct answer, but when I multiply by the conjugate from the denominator it doesn't come out to this.
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NO...the bottom is r² ( 1 - r² / 4 ) - 1 = ( r² - 2)² / [-4] ; top will be [ - r^3 / 2 ] [r √( 1 - r² / 4 ) + 1 ]
(-1/2 * r^3)/(r * sqrt(1 - (1/4 * r^2)) - 1) =
-½ r³
----------------------- =
r √(1 - (¼ r²)) - 1
-½ r³............................r √(1 - (¼ r²)) + 1
----------------------- • ------------------------- =
r √(1 - (¼ r²)) - 1........r √(1 - (¼ r²)) + 1
-½ r³ r √(1 - (¼ r²)) + -½ r³
----------------------------------- =
(r √(1 - (¼ r²)))² - 1
-½ r⁴ √(1 - (¼ r²)) + -½ r³
----------------------------------- =
r² (1 - (¼ r²)) - 1
-½ r³ [r √(1 - (¼ r²)) + 1]
-------------------------------- =
r² [1 - (¼ r²)) - 1]
-½ r³ [r √(1 - (¼ r²)) + 1]
-------------------------------- =
r² (¼ r²))
-½ r [r √(1 - (¼ r²)) + 1]
-------------------------------- =
(¼ r²))
-2 r [r √(1 - (¼ r²)) + 1]
-------------------------------- =
r²
-2 [r √(1 - (¼ r²)) + 1]
-------------------------------- =
r
-2 r √(1 - (¼ r²)) - 2
-------------------------------- =
r
-2 √(1 - (¼ r²)) - 2 =
-2 (√(1 - (¼ r²))+1).........ANS
// Unless I made a mistake, I believe the answer should be NEGATIVE
// and not POSITIVE as you've written above