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Rectifier
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Is ## "\frac{0}{\infty}"=0 ## ?
No, it is undefined because you are treating infinity as though it was a number. It isn't. You can't, in most circumstances, use infinity in normal math and expect meaningful results.Rectifier said:Is ## "\frac{0}{\infty}"=0 ## ?
## \infty ## isn't a number, so you shouldn't be dividing by it.Rectifier said:Is ## "\frac{0}{\infty}"=0 ## ?
No, it's still undefined in most contexts, I think. What IS true is that as n approaches infinity, 0/n approaches 0..Scott said:## \infty ## isn't a number, so you shouldn't be dividing by it.
That said, in most contexts, yes, ## \frac{0}{\infty}=0 ##
OK, in most meaningful contexts.phinds said:No, it's still undefined in most contexts, I think. What IS true is that as n approaches infinity, 0/n approaches 0.
phinds said:What contexts did you have in mind?
That last limit is 0. Don't write it as ##\frac 0 {\infty}##, either with or without quotes.Rectifier said:I was trying to calculate following limit:
## \lim_{x\rightarrow \infty} \frac{x^4 + x \ln x}{x + \left( \frac{2}{3} \right)^x} = \lim_{x\rightarrow \infty} \frac{x^4 \left( 1 + \frac{\ln x}{x^3}\right) }{ x \left( 1 + \frac{ \left( \frac{2}{3} \right)^x }{x} \right) } = \lim_{x\rightarrow \infty} x^3\frac{1 + \frac{\ln x}{x^3} }{ 1 + \frac{ \left( \frac{2}{3} \right)^x }{x} } ##
## \lim_{x\rightarrow \infty} x^3 = \infty ##
## \lim_{x\rightarrow \infty}\frac{\ln x}{x^3}=0 ##
## \lim_{x\rightarrow \infty} \left( \frac{2}{3} \right)^x = 0 ##
## \lim_{x\rightarrow \infty} \frac{ \left( \frac{2}{3} \right)^x }{x}
= "\frac{0}{\infty}"##
Rectifier said:Is ## "\frac{0}{\infty}"=0 ## ?
That approach is much better, because 0*0 is defined in the real numbers.Rectifier said:I think I got it.## \lim_{x\rightarrow \infty} \frac{ \left( \frac{2}{3} \right)^x }{x}
= \lim_{x\rightarrow \infty} \frac{ 1 }{x} \left( \frac{2}{3} \right)^x ##
## \lim_{x\rightarrow \infty} \frac{ 1 }{x} = 0 ##
## \lim_{x\rightarrow \infty} \left( \frac{2}{3} \right)^x = 0 ##
phinds said:No, it's still undefined in most contexts, I think. What IS true is that as n approaches infinity, 0/n approaches 0.
EDIT: I hasten to add, I don't do any math where there IS any meaningful context for 0/infinity, so I could be wrong. What contexts did you have in mind?
mfb said:That approach is much better, because 0*0 is defined in the real numbers.
Why do you think so? 0 is a number, and multiplying any number by 0 is 0.micromass said:0*0 is undefined?
Huh? I said it is defined.micromass said:0*0 is undefined?
micromass, than you for that correction, but I wonder if you could expand on it for me a bit? I'm confused as to how it is that "0/infinity = 0" is not treating infinity like a normal numbermicromass said:You are wrong. In most contexts where the OP makes sense, it is indeed true that ##0/\infty = 0##.
But that's my point. I've always been told that you CAN'T normally treat infinity like a normal number. Is this wrong?mfb said:If you treat infinity like a normal number, you should be allowed to multiply by it, but then you get ##0=0\cdot \infty## which is not well-defined any more.
Correct. You can still perform some limited operations in most contexts, and 0/infinity=0 is one of them (e.g. for limits). That is not treating it as normal number, that is a specific rule for this operation.phinds said:I've always been told that you CAN'T normally treat infinity like a normal number.
OK, thanks.mfb said:Correct. You can still perform some limited operations in most contexts, and 0/infinity=0 is one of them (e.g. for limits). That is not treating it as normal number, that is a specific rule for this operation.
mfb said:Huh? I said it is defined.
Daymare said:Why is it that we don't define operations like 0*infinity and 0/infinity etc.They do occur in maths.Is it because defining these things causes inconsistencies?
Daymare said:You could define it as 0
In the first example, the limit is that of a sequence of finite length lines becoming longer without any limit so the result is a limit on 0.n=0 as n becomes infinite. In cases like this you must always consider the underlying sequence that leads to the expression.micromass said:Right, and this is done in measure theory, where you rigorously define what an area is. The logic is that a straight line has length ##\infty## and width ##0## and has an area of ##0##. So if you define ##0\cdot \infty = 0## then this is consistent.
But why is this not done in the rest of mathematics. Consider ##f(n) = 1/n## and ##g(n) = n^2##. Then ##f(n)\rightarrow 0## and ##g(n)\rightarrow +\infty##. But ##f(n)g(n)\rightarrow +\infty##. So in this sense at least, ##0\cdot \infty = \infty## also makes sense.
And if ##f(n) = 1/n## and ##g(n) = n##, then ##f(n)g(n)\rightarrow 1##. So ##0\cdot \infty = 1## also makes sense in this sense.
Infinity is a concept in mathematics that represents a quantity or value that is unbounded, limitless, or never-ending. It is often symbolized by the infinity symbol (∞) and is used in various mathematical operations and equations.
No, infinity is not a tangible or attainable value. It is a theoretical concept used in mathematics to represent a concept of boundlessness or endlessness.
The concept of infinity in limits is used to describe the behavior of a function or sequence as its input or values approach infinity. It helps determine the ultimate or maximum value of a function or sequence as its inputs become larger and larger.
Yes, there are different types of infinity in mathematics. For example, there is countable infinity, which is used to describe the number of counting numbers (1, 2, 3, ...). There is also uncountable infinity, which is used to describe the number of real numbers on a number line.
The relationship between infinity and limits is that infinity is often used in limits to describe the behavior of a function or sequence as its inputs approach infinity. In other words, infinity is used to represent the ultimate or maximum value of a function or sequence as its inputs become larger and larger.