What Is Infinity? Analyzing the Limit of x→∞

[z.js]

What is ∞? I know it means infinity, but consider this:
$$\lim_{x\rightarrow +\infty} \frac{x^2 + 2x + 1}{x+1}$$
Numerator: $\displaystyle \lim_{x\rightarrow \infty} x^2 + 2x + 1$ $$= \infty$$
Denominator: $\displaystyle \lim_{x\rightarrow \infty} x + 1$ $$= \infty$$
The numerator's ∞ is "bigger" than the denominator's, and the fraction tends to $\frac{∞}{∞}$ but I do NOT think it is equal to $1$. Since it is $\frac{big \infty}{small \infty}$, then it would $= ∞$.
And more...
$$\lim_{x\rightarrow +\infty} \frac{x^2 + 2x + 1}{x+1} = \lim_{x\rightarrow +\infty} x+1 = ∞$$
This is the same as $\frac{big \infty}{small \infty} = \infty$. Doesn't this prove that the numerator's $∞$ is bigger than the denominator's?

$$\lim_{x\rightarrow +\infty} \frac{x^2 + 2x + 1}{x+1} = \lim_{x\rightarrow +\infty} \frac{x^2 + 2x + 1}{x+1}$$
$$\frac{big \infty}{small \infty} = \lim_{x\rightarrow +\infty}\frac{(x + 1)^2}{(x + 1)}$$
$$\infty = \lim_{x\rightarrow +\infty} x + 1$$
$$\infty = \infty$$


Also, what is $$\frac{3\infty}{2\infty} ?$$

 

[stfz]

infinity is absorbent (I think that is the term). So INF + 3 = INF, 3*INF = INF, INF/2 = INF,, so on.

 

[z.js]

Well does my equation make sense?
$∞$ might be absorbent, but I'm not sure.

 

[gopher_p]

https://www.physicsforums.com/showthread.php?t=507003

The infinity of calculus is basically just shorthand for "grows without bound". You can't really do arithmetic with it, even though sometimes it looks like you can.

So when we write $\lim_{x\rightarrow\infty}f(x)=\infty$ and say "The limit as x approaches infinity of f of x equals infinty", what we really mean is "as x grows without bound, so does f of x".

There are no actual equations involving infinity in calculus, even though, again, the notation makes it look like there are.

 

[phinds]

What is ∞? ...

One thing I would point out is that you speak as though there was only a single infinity, but that's not true. There are an infinity of DIFFERENT infinities (strictly speaking, these are the cardinalities of sets), each bigger than the other. Look up "Alepha Null" for more.

 

[micromass]

One thing I would point out is that you speak as though there was only a single infinity, but that's not true. There are an infinity of DIFFERENT infinities (strictly speaking, these are the cardinalities of sets), each bigger than the other. Look up "Alepha Null" for more.

This is totally irrelevant to the OP. The infinity for the OP is the infinity for limits. For example, you have things like

$$\lim_{x\rightarrow a} f(x) = +\infty$$

These kind of infinities are just symbols but they can be given actual existence by the extended real line $$\overline{\mathbb{R}} = \mathbb{R}\cup \{-\infty,+\infty\}$$. In this sense, there are only two infinities: minus and plus infinity.

Cardinalities of sets and aleph null have nothing at all to do with this.

 

[phinds]

This is totally irrelevant to the OP.

OK, good point.

 

[Edgardo]

Here is a definition of what the limit means when it involves infinity:

$$\lim_{x \to +\infty}f(x) = +\infty$$
if for every number M>0 there is a corresponding number N such that
$f(x)>M$ whenever $x>N$.

Intuitively this means, if I give you a positive number M, then you can find a number N such that
$x>N$ implies $f(x)>M$.

References:
http://www.ocf.berkeley.edu/~yosenl/math/epsilon-delta.pdf
http://www.math.oregonstate.edu/hom...tStudyGuides/SandS/lHopital/define_limit.html

See these videos: Example 1, example 2.

 

[z.js]

One thing I would point out is that you speak as though there was only a single infinity, but that's not true. There are an infinity of DIFFERENT infinities (strictly speaking, these are the cardinalities of sets), each bigger than the other. Look up "Alepha Null" for more.

Why, sure as you live, that's JUST IT!

 

[z.js]

But, what about my equations?

 

[Mark44]

But, what about my equations?

What is ∞? I know it means infinity, but consider this:
$$\lim_{x\rightarrow +\infty} \frac{x^2 + 2x + 1}{x+1}$$
Numerator: $\displaystyle \lim_{x\rightarrow \infty} x^2 + 2x + 1$ $$= \infty$$
Denominator: $\displaystyle \lim_{x\rightarrow \infty} x + 1$ $$= \infty$$
The numerator's ∞ is "bigger" than the denominator's, and the fraction tends to $\frac{∞}{∞}$ but I do NOT think it is equal to $1$. Since it is $\frac{big \infty}{small \infty}$, then it would $= ∞$.
And more...
$$\lim_{x\rightarrow +\infty} \frac{x^2 + 2x + 1}{x+1} = \lim_{x\rightarrow +\infty} x+1 = ∞$$
This is the same as $\frac{big \infty}{small \infty} = \infty$. Doesn't this prove that the numerator's $∞$ is bigger than the denominator's?

No. "Big infinity" and "small infinity" don't make much sense here. This limit has the form $[\frac{\infty}{\infty}]$. What I wrote is notation for one indeterminant form. There are others.

$$\lim_{x \to \infty} \frac{x^2 + 2x + 1}{x + 1} = \lim_{x \to \infty} \frac{(x + 1)^2}{x + 1}$$
$$= \lim_{x \to \infty} x + 1 = \infty$$
That's all you need to say. The fraction that I cancelled, (x + 1)/(x + 1) is always equal to 1 for any value of x other than -1, so the value is still 1 as x grows large without bound.

$$\lim_{x\rightarrow +\infty} \frac{x^2 + 2x + 1}{x+1} = \lim_{x\rightarrow +\infty} \frac{x^2 + 2x + 1}{x+1}$$
$$\frac{big \infty}{small \infty} = \lim_{x\rightarrow +\infty}\frac{(x + 1)^2}{(x + 1)}$$
$$\infty = \lim_{x\rightarrow +\infty} x + 1$$
$$\infty = \infty$$


Also, what is $$\frac{3\infty}{2\infty} ?$$

We don't do arithmetic operations on ∞. This limit, though, is similar to what you're asking.
$$ \lim_{x \to \infty} \frac{3x}{2x} = \lim_{x \to \infty} \frac{x}{x} \frac{3}{2} = \frac 3 2$$
In the last limit expression, x/x is always 1 for any value of x other than 0, so its limit is also 1 as x grows large. That leaves us with 3/2 for the limit.