A question on a specific limit

[Agent Smith]

This was taught to me:

$\displaystyle \displaystyle \lim_{x \to 0} \frac{1}{x} = \infty$ (A)

1. A graph or a table will quasi-verify the equality.
2. y = 0 or the y axis is the vertical asymptote

But $\displaystyle \frac{1}{0} \ne \infty$, it's undefined (B)

What's the difference between

$\displaystyle \displaystyle \lim_{x \to 0} \frac{1}{x}$ (A) and $\displaystyle \frac{1}{0}$ (B)?

 

[mario99]

99% of Agent Smith's questions are about infinity.



And here is the ambiguous question #88888888888888888889

 

[Agent Smith]

99% of Agent Smith's questions are about infinity.



And here is the ambiguous question #88888888888888888889

Glad that you found that funny.

 

[Agent Smith]

I mean x = 0 is the vertical asymptote.

 

[mario99]

$\displaystyle \lim_{x\rightarrow 0} \frac{1}{x} \neq \infty$

$\displaystyle \lim_{x\rightarrow 0} \frac{1}{x} = \text{DNE}$

DNE means does not exist.

 

[Agent Smith]

$\displaystyle \lim_{x\rightarrow 0} \frac{1}{x} \neq \infty$

$\displaystyle \lim_{x\rightarrow 0} \frac{1}{x} = \text{DNE}$

DNE means does not exist.

Oh great! This is one of the issues that's non liquet to me.

When $\displaystyle \lim_{x \to c} f(x) = \infty$, the following ways of expressing this are seen:
1. The limit is unbounded
2. The limit is infinity
3. The limit does not exist
Can you explain to me each expression and how they're all equivalent?

Gracias

 

[Agent Smith]

Another thing, I was supposed to have asked the question below and its related to the original question in my OP.

$\displaystyle \displaystyle \lim_{x \to \infty} \frac{1}{x} = 0$

Does that mean then that $\frac{1}{\infty} = 0$

If yes, then after some algebraically valid manipulations we get $\frac{1}{0} = \infty$. Mind you this isn't a completely nonsensical answer as $\frac{1}{0} = \text{Sunday}$. Brahmagupta, the ancient Hindu mathematician, thought that a number divided by zero = infinity.

 

[mario99]

Oh great! This is one of the issues that's non liquet to me.

When $\displaystyle \lim_{x \to c} f(x) = \infty$, the following ways of expressing this are seen:
1. The limit is unbounded
2. The limit is infinity
3. The limit does not exist
Can you explain to me each expression and how they're all equivalent?

Gracias

if $\displaystyle \lim_{x\rightarrow c^{+}} f(x) \neq \lim_{x\rightarrow c^{-}} f(x)$ then $\displaystyle \lim_{x\rightarrow c} f(x) = \text{DNE}$

Unbounded and $\infty$ $(\text{or}-\infty)$ are the same thing.

When $\displaystyle \lim_{x\rightarrow c^{+}} f(x) = \infty$
And $\displaystyle \lim_{x\rightarrow c^{-}} f(x) = \infty$
Then $\displaystyle \lim_{x\rightarrow c} f(x) = \infty$

Another thing, I was supposed to have asked the question below and its related to the original question in my OP.

$\displaystyle \displaystyle \lim_{x \to \infty} \frac{1}{x} = 0$

Does that mean then that $\frac{1}{\infty} = 0$

If yes, then after some algebraically valid manipulations we get $\frac{1}{0} = \infty$. Mind you this isn't a completely nonsensical answer as $\frac{1}{0} = \text{Sunday}$. Brahmagupta, the ancient Hindu mathematician, thought that a number divided by zero = infinity.

$\displaystyle \frac{1}{\infty}$ has no meaning

Also

$\displaystyle \frac{1}{0}$ has no meaning

But

$\displaystyle \lim_{x\rightarrow \infty} f(x) = \frac{1}{\infty} = 0$ has meaning

Also

$\displaystyle \lim_{x\rightarrow 0} f(x) = \frac{1}{0} = \infty$ has meaning

 

[Agent Smith]

Ok @mario99 , what's the difference then between, it boils down to, $\displaystyle \displaystyle \lim_{x \to \infty} \frac{1}{x}$ and $\displaystyle \frac{1}{\infty}$?

I'll explain my issue with this, that the former is defined and the latter is not:

The only difference between the two is $\displaystyle \displaystyle \lim_{x \to \infty}$. We know that if we were to construct a table, if $\displaystyle x = 3$ then we'd have to compute $\displaystyle \frac{1}{3}$. How does then $\displaystyle \displaystyle \lim_{x \to \infty}$ confer meaning to an otherwise meaningless $\displaystyle \frac{1}{\infty}$.

I know the consensus is $\displaystyle \frac{1}{0}$ and, according to you, $\displaystyle \frac{1}{\infty}$ is meaningless, but consider this a scenario of limited options and meaningless is not a choice available to you. Would you rather say that $\displaystyle \frac{1}{0} = 3$ or would you prefer $\displaystyle \frac{1}{0} = \infty$?

Apologies if this comes off as pseudomathematical crankery. Trying to comprehend advanced mathemetical concepts I'm probably not ready for and these are the questions that enter my mind at the moment.

 

[mario99]

Ok @mario99 , what's the difference then between, it boils down to, $\displaystyle \displaystyle \lim_{x \to \infty} \frac{1}{x}$ and $\displaystyle \frac{1}{\infty}$?

Everything is different between them!
One difference is that the first is zero while the second is an invalid expression.
The first says if x goes to infinity, y goes to zero. The second says nothing.
The first is a limit of a function. The second is a meaningless fraction, or a fraction of nothing.

$1$ is a number while $\infty$ is not a number (I don't even know what it is!).
How can you divide a number by a weird thing such $\infty$? $\displaystyle \left(\frac{1}{\infty}\right)$
It's like dividing apples by bananas!

I have already told you that $\displaystyle \frac{1}{\infty} \ \ \text{and} \ \ \frac{1}{0}$ are meaningless without a limit. But you insist that you want to use them in Algebraic Manner such as $\displaystyle \frac{1}{0} = \infty$ then $\displaystyle \frac{1}{\infty} = 0$ then $\displaystyle 1 = 0 \times \infty$ then $\displaystyle 1 = 0$ or $\displaystyle 1 = \infty$. But all these expressions are invalid without a limit. If you keep arguing in such thing, it's like you're running in circles. You will go no where.

I think that you are not a student. Maybe you are a professor or a scientist of some kind. We are normal people, so please lower down your math level in such a way that we can understand your questions!

 

[Agent Smith]

Everything is different between them!
One difference is that the first is zero while the second is an invalid expression.
The first says if x goes to infinity, y goes to zero. The second says nothing.
The first is a limit of a function. The second is a meaningless fraction, or a fraction of nothing.

$1$ is a number while $\infty$ is not a number (I don't even know what it is!).
How can you divide a number by a weird thing such $\infty$? $\displaystyle \left(\frac{1}{\infty}\right)$
It's like dividing apples by bananas!

I have already told you that $\displaystyle \frac{1}{\infty} \ \ \text{and} \ \ \frac{1}{0}$ are meaningless without a limit. But you insist that you want to use them in Algebraic Manner such as $\displaystyle \frac{1}{0} = \infty$ then $\displaystyle \frac{1}{\infty} = 0$ then $\displaystyle 1 = 0 \times \infty$ then $\displaystyle 1 = 0$ or $\displaystyle 1 = \infty$. But all these expressions are invalid without a limit. If you keep arguing in such thing, it's like you're running in circles. You will go no where.

I think that you are not a student. Maybe you are a professor or a scientist of some kind. We are normal people, so please lower down your math level in such a way that we can understand your questions!

I kinda understand what you might be getting at. This $\displaystyle \displaystyle \lim_{x \to \infty} \frac{1}{x} = 0$ is not saying $\displaystyle \frac{1}{\infty} = 0$. Now I also recall the statement "a limit has to be finite" and now I'm confused all over again. $\displaystyle \displaystyle \lim_{x \to \infty} \frac{1}{x} = 0$, notice "$\displaystyle \lim$" in the expression. Isn't this exactly what I was referring to above? The notation, perhaps something else, doesn't allow us to express "there's no limit" and so we have to phrase it as e.g. "the limit is unbounded/infinity".

 

[Agent Smith]

Source: Wikipedia
Apologies for the error
However are $\displaystyle \frac{0}{0}$ and $\displaystyle \frac{a}{0}, a \ne 0$ logically connected?

 

[Agent Smith]

Also how does the statement for $\displaystyle \displaystyle \lim_{x \to 0} \frac{1}{x} = \infty$, that "the limit DNE" square with set theory's axiom of infinity. One is apparently saying infinity does not exist, and the other is assuming it does.

 

[khansaheb]

that "the limit DNE"

That does NOT mean that "infinity" does not exist - it only states that the LIMIT does not exist (or the limit cannot be calculated).

 

[Agent Smith]

That does NOT mean that "infinity" does not exist - it only states that the LIMIT does not exist (or the limit cannot be calculated).

Ok. Gracias to you and @mario99 . Very kind of you to assist gentlemen/ladies.
A limit has to be finite. It would be a contradiction to say the limit is unlimited (infinity).