# Thread: Positive or Negative Infinity?

1. ## Positive or Negative Infinity?

$\displaystyle \lim_{x -> 0} (5x - \frac{4}{x})$

Is the answer to this question positive or negative infinity? The calculator at Online Limit Calculator gives positive infinity, but wouldn't -4/0 be negative?

2. Originally Posted by RogueDemon
$\displaystyle \lim_{x -> 0} (5x - \frac{4}{x})$ Is the answer to this question positive or negative infinity? The calculator at Online Limit Calculator gives positive infinity, but wouldn't -4/0 be negative?
Have a look at this.
It appears that it depends on the side of 0 we choose.

3. Originally Posted by RogueDemon
$\displaystyle \lim_{x -> 0} (5x - \frac{4}{x})$

Is the answer to this question positive or negative infinity? The calculator at Online Limit Calculator gives positive infinity, but wouldn't -4/0 be negative?

The calculator is correct. Take into account that $\displaystyle \infty\neq +\infty$ . Look at the three posibilities: two sided, plus and minus.

4. I don't understand why $\displaystyle \infty \neq +\infty$.

5. Originally Posted by NOX Andrew
I don't understand why $\displaystyle \infty \neq +\infty$.
Supposing $\displaystyle f$ defined in a punctured neighbourhood of $\displaystyle a$:

$\displaystyle \displaystyle\lim_{x \to a}{f(x)}=+\infty \Leftrightarrow \forall{M>0}\;\exists{ \epsilon >0}:\textrm{\;if\;}0<|x-a|< \epsilon \textrm{\;then\;} f(x)>M$

$\displaystyle \displaystyle\lim_{x \to a}{f(x)}=\infty \Leftrightarrow \forall{M>0}\;\exists{ \epsilon >0}:\textrm{\;if\;}0<|x-a|< \epsilon \textrm{\;then\;}| f(x)|>M$

Our case $\displaystyle |f(x)|=|5x-4/x|\to +\infty$ as $\displaystyle x\to 0$ (two sided) so, satisfies the second definition.

6. Strictly speaking, that limit does not exist, even as "plus infinity" or "negative infinity"! The two one sided limits are
$\displaystyle \lim_{x\to 0^+} f(x)= -\infty$ (for example, if x= 0.0001, $\displaystyle f(x)= f(0.0001)$$\displaystyle = 5(.0001)- \frac{4}{.0001}$$\displaystyle = .0005- 40000= -39999.9995$ and
$\displaystyle \lim_{x\to 0^-}f(x)= +\infty$ (for example, if x= -0.0001, $\displaystyle f(x)= f(-0.0001)= 5(-.0001)- \frac{4}{-.0001}$[tex]= -.0005+ 40000= 39999.9995.

I don't understand why $\displaystyle -\infty\ne\infty$.
I was sorely tempted to ask if you understood why $\displaystyle -5\ne 5$!

Of course, neither "$\displaystyle \infty$" nor "$\displaystyle -\infty$" is actually a number. We are really talking about limits of functions or sequences. $\displaystyle -\infty$ and $\displaystyle \infty$ are different because sequence that converge to each, such as 1, 2, 3, 4, ..., n, ..., which has limit $\displaystyle \infty$, and -1, -2, -3, -4, ..., -n, ..., which has limit $\displaystyle -\infty$, are clearly getting farther apart so cannot converge to the same thing.

(From a "topological" viewpoint, adding $\displaystyle \infty$ and $\displaystyle -\infty$ "compactifies" the real number line making it topologically equivalent to a closed interval. Given any topological space, we can always find its "one point compactification", adding a single point, and "open" sets containing that one point, to make it compact- However, it changes the topological properties of the space much more than the "Stone-Chech compactification" above. If we did that with the set of real numbers, we would get a single "point at infinity" but now the it would be topologically equivalent to a circle rather than a line segment.)

7. You seem to have misquoted me. I understand the difference between negative infinity and infinity, just not the difference between positive infinity and infinity.

8. Originally Posted by HallsofIvy
Strictly speaking, that limit does not exist, even as "plus infinity" or "negative infinity"!
When we compactify $\displaystyle \mathbb{C}$ adding only one point to obtain an homeomorphism with $\displaystyle S^2$ , we denote the added point to $\displaystyle \mathbb{C}$ by $\displaystyle \infty$ .

When we compactify $\displaystyle \mathbb{R}$ adding only one point to obtain an homeomorphism with $\displaystyle S^1$, how do you denote the added point?

,

,

,

,

,

,

,

### difference between plus and minus infinity

Click on a term to search for related topics.