Question

f(x) = |x|/x, if x =/= o

0, if x = 0

@dan815

Answer 1

*gotta check continuity

Answer 2

f(x) is continuous at x=a if the limit exists and equals f(a)

Answer 3

Haha actually, my question was why my textbook stated |x|/x as -1 xD

Answer 4

I might have called my friend and told him |x| when x -> 0 approaches from the left is -1.. now he's all confused thanks to me

Answer 5

Notice when x < 0, we have |x| = -x\[\lim\limits_{x\to 0^{\color{red}{-}}} ~\dfrac{|x|}{x} = \lim\limits_{x\to 0^{\color{red}{-}}} ~\dfrac{-x}{x} = ?\]

Answer 6

Yeah but why take |x| as -x?

Answer 7

thats the definition of absolute value function :

\[|x| = \left\{\begin{array}{} -x&:&x\lt 0\\x&:&x\ge 0\end{array}\right.\]

Answer 8

Intuitively, the key is that x < 0 so x is negative. Therefore, - x > 0 or positive as it should be for an absolute value.

Answer 9

OH Okay okay okay I got it xD

Answer 10

so we write |x| when limit of x-> 0 from the right as |x|/x = x/x = 1

Answer 11

Yes! since left and right limits dont agree, the limit as x->0 itself doesn't exist.

Answer 12

Left hand limit =/= Right hand limit, so it is discontinous at x = 0

Answer 13

That has to mentioned everytime LHL and RHL do not match up?

Answer 14

thats right, but just keep in mind that for f(x) to be continuous at x=a, we must have :

LHL = RHL = f(a)

Answer 15

LHL = RHL is not a sufficient condition for continuity

the limit must equal the function value also

Answer 16

f(a) -> f(0) = 0 but they all still don't match up, so yeah. I got another question, I'll close this one, more medals for you (: