lim x- 0 to (|5x-6| - |5x+6|)/x? using L hspitals rule i can figure out that it's -10 but how do i do this without using derivatives?

Question

lim x-> 0 to (|5x-6| - |5x+6|)/x? using L hspitals rule i can figure out that it's -10 but how do i do this without using derivatives?

anonymous · Answer

l hspitals rule is derivatives

amistre64 · Answer

youd might have to split it up into parts

amistre64 · Answer

$$\lim_{x->0} \frac{|5x-6| - |5x+6|}{x}$$

   (5x-6)/x
 -(5x+6)/x
---------
      -12/x

  (-5x+6)/x
- (5x+6)/x
----------
     -10x/x = -10

   (5x-6)/x
-(-5x-6)/x
---------
      10x/x = 10

seem to be all the possible scenarios right?

zarkon · Answer

for x near zero  |5x-6|=-(5x-6) and |5x+6|=5x+6

amistre64 · Answer

maybe also opp-opp

 -(5x-6)/x
 -(5x+6)/x
----------
     -10x/x = -10

amistre64 · Answer

i loathe absolute values :)

zarkon · Answer

$$\lim_{x->0} \frac{|5x-6| - |5x+6|}{x}=\lim_{x->0} \frac{-(5x-6) - (5x+6)}{x}$$

$$=\lim_{x->0} \frac{-5x+6 - 5x-6}{x}=\lim_{x->0} \frac{-10x}{x}=-10$$