Why is -1/(0²) considered -Infinity while 1/0 is undefined?

[cscott]

Why does my calculator tell me -1/(0^2) = -infinity. How is this different from 1/0?

 

[Euclid]

1/0 isn't anything, since 1/x approaches infinity if x --> 0 from the right (x>0) and 1/x approaches -infinity if x --> 0 from the left (x<0).
By contrast, -1/x^2 is always negative, so it approaches negative infinity as x --> 0 no matter which direction you come from.
Bear in mind, I'm not saying that -1/0^2 equals - infinity. It's not really defined actually.
What kind of calculator are you using anyway? Mine always says "error - divide by 0" if I put in 1/0.

 

[dav2008]

I think his question is why his calculator (TI-89) says 1/0 is undefined but 1/0² says infinity.

It just has to do with how the calculator calculates things I guess.

Edit: I tried it for other powers and it seems like a/0ⁿ is given as infinity for even n and "undef" for odd n if n is positive.

If n is negative it gives a/0ⁿ as 0. If n is zero it gives it as a, but writes a warning message saying that 0⁰ was replaced by 1.Edit: I think latex is broken...

 

[cscott]

Alright, thanks.

...and I agree, Latex is broken.

 

[TD]

Never trust your calculator for mathematical 'subtleties' like this, use your mind

 

[Hurkyl]

Incidnetally, the function

-1/x²

does have a continuous extension to x=0 in the extended real numbers. Whereas

-1/x

does not.

 

[dav2008]

Incidnetally, the function

-1/x²

does have a continuous extension to x=0 in the extended real numbers. Whereas

-1/x

does not.

That makes sense. The calculator is probably computing the limit of those functions as they go to 0, which is infinity for even functions and undefined for odd.