I confronted something in math which is way too confusing to me

[sankalpmittal]

I confronted "something" in math which is way too confusing to me !

Hello forums ,
I am currently in class 10th , and 15 years... I was doing Khan Academy Pre-calculus yesterday and I confronted something too weird ... not weird though but way too confusing...


Here is the link :
http://www.khanacademy.org/video/more-limits?playlist=Precalculus

Now what I am confused with , is that , Sal ( That's not a name of a tree !) says that

lim (x-2|x|)/|x|
x→0

Then he says that when limit x tends to 0 from positive side then :

lim (x-2|x|)/|x|
x→0⁺

is SAME AS

lim (x-2x)/x
x→0⁺

which is equal to

lim (x-2|x|)/|x| = -1
x→0⁺

This makes sense to me and here comes which confuses me :

Now Sal says that when x approaches 0 from negative side then :

lim (x-2|x|)/|x|
x→0^-

is SAME AS

lim (x-2(-x))/-x
x→0^-

which is equal to

lim (3x)/-x
x→0^-

SO

lim (x-2|x|)/|x| = -3
x→0^-

This does not make sense ! |-x| → |x| right ?
Why he wrote |-x| = -x ?! OR |x| = -x ? ! where x < 0 ?!

Can someone explain it to me , please ?

Thanks in advance ...

 

[yenchin]

|x| is the absolute value of x, so you "ignore the minus sign". For positive number, an example would be say, |5|=5. So you can see that |x|=x if x > 0. For negative x, say -3, its absolute value is |-3|=3, yes? But 3=-(-3), so |x|= -x if x < 0.

 

[jedishrfu]

As Yenchin says you can't work with |x| directly and so you break the problem up into two cases:
- one equation where x is positive so that you can replace |x| with x
- one equation where x is negative so that you can replace |x| with -x

these replacements come from the definition of the absolute value function:
- for x>0 the value is x
- for x=0 the value is 0
- for x<0 then value is -x

and since x=0 can't be used due to division by zero being an undefined operation.

 

[sankalpmittal]

|x| is the absolute value of x, so you "ignore the minus sign". For positive number, an example would be say, |5|=5. So you can see that |x|=x if x > 0. For negative x, say -3, its absolute value is |-3|=3, yes? But 3=-(-3), so |x|= -x if x < 0.

As Yenchin says you can't work with |x| directly and so you break the problem up into two cases:
- one equation where x is positive so that you can replace |x| with x
- one equation where x is negative so that you can replace |x| with -x

these replacements come from the definition of the absolute value function:
- for x>0 the value is x
- for x=0 the value is 0
- for x<0 then value is -x

and since x=0 can't be used due to division by zero being an undefined operation.

Thanks yenchin and jedishrfu !
Now I know why Sal wrote |x| = -x , where x<0 !
You two mean that here x < 0 and so x is negative.
So -x will here be positive i.e. -x>0 or x<0
As we know that "absolute function || " yields positive value , so |x| = -x where x<0
AND
|x| = x , where x>0
AND
|-x| = x , where x>0
AND
|-x| = -x , where x<0 , in number form I write it like -: x=-1
|-(-1)| = |1| = 1 which is same as -x right ?
Is this what you two mean right ?

Now I can move forward in pre-calculus.

Thanks a lot !

 

[Mark44]

Thanks yenchin and jedishrfu !
Now I know why Sal wrote |x| = -x , where x<0 !
You two mean that here x < 0 and so x is negative.
So -x will here be positive i.e. -x>0 or x<0
As we know that "absolute function || " yields positive value , so |x| = -x where x<0
AND
|x| = x , where x>0
AND
|-x| = x , where x>0
AND
|-x| = -x , where x<0 , in number form I write it like -: x=-1
|-(-1)| = |1| = 1 which is same as -x right ?
Is this what you two mean right ?

Now I can move forward in pre-calculus.

Thanks a lot !

What you have above is sort of the definition of the absolute value. In a briefer form, it is:
|x| = x, if x >= 0
|x| = -x, if x < 0