Limit question limit (x->0) 1/|x|

[booya123]

limit (x->0) 1/|x|

 

[AdrianZ]

the limit doesn't exist.
lim (x-> o+) = + ∞
lim (x-> o-) = - ∞

so limit of the function at the point zero is not existent by definition.

 

[Mentallic]

the limit doesn't exist.
lim (x-> o+) = + ∞
lim (x-> o-) = - ∞

so limit of the function at the point zero is not existent by definition.

Not true. The function is 1/|x| not 1/x

 

[AdrianZ]

Not true. The function is 1/|x| not 1/x

aah yea you're right. didn't notice the abs function there. anyway, when x approaches zero from left (and right) the function approaches +∞. so the limit is +∞.

that means for all positive deltas and M's if |x - 0| < delta, then f(x) > M. for any large M you like. intuitively you can say: the larger you want M to be, the smaller you need to take delta.

 

[HallsofIvy]

Which still means that there is no limit. Saying "$\lim f(x)= \infty$" is just a way of saying that the limit does not exist for a particular reason.

 

[AdrianZ]

well, I think if we're working on extended real number line then it doesn't hurt to say the limit of the function is +∞ because +∞ ∈ R*. and plus the epsilon, delta definition works fine with this example. correct me if I'm wrong.