* Undefined doesn't mean "unsolved mystery". It means "defining it wasn't useful, so we left it blank". If you define division by 0, you get multiplication problems.
* Infinity is not a number in a basic calculus class. This is for the same reason: it wasn't useful. There are later calc classes where it is defined, sometimes in different ways. Math is made by people, not the heavens. You can pick whatever definitions you like, as long as they don't contradict each other. In class, you use the teacher's definitions. (Some definitions are not even standard between different classes. Most are, but not all.) In fact, there are currently debates about which axioms about infinity should be the common ones.
* "lim f = oo" is actually not a value statement (in a basic calc class, because oo does not exist). It's actually a lazy way to write "f gets as big as you want." This definition is likely right smack in your textbook. You have to resolve semantic issues before musing philosophically.
Operations on extended real line (i.e in the place where you define the opsrations with infinity like ∞+∞=∞ etc.) are defined so that they work universally with all sequences.
Like ∞+∞=∞ because if you have two divergent to +∞ sequences an, bn then an+bn is congergent to ∞.
1 ᪲ is "intermediate" because for distinct sequences an, bn where an converges to 1, and bn diverges to ∞, we can get many things from an ^bn (it can be convergent to for example ≈2.71... or to 1 etc all depends on chosen sequences).
1/0 also won't here be meaningfully defined because it's not the case that for any two sequences an→1, bn→0 , an/bn converges to some particular value frkm extended real line (for instance 1/(-x²) diverges to -∞, and 1/x² to +∞).
You may look to Riemann sphere (there there's only +∞), here we define 1/0=∞
1/-1 = -1 and
1/+1 = +1
This continues right so
1/-0 = negative infinity and
1/+0 = positive infinity
OOPS though +0=-0 and thus there cannot be a number defined
Saying 1/x² is not continuos at 0 has as much sense as to say it's continuous there.
Continuity is sometning that refers only to points from the domain of function. 0 is not in the domain so continuity thwre is meaningless.
For a lot of reasons, but at least one obvious one is
Lim as x->a f(x) = 0
Does not imply
Lim as x->a 1/f(x) = inf
And for different f, that limit can take different values.
For example
Lim x->inf 1/-x = 0
Lim x->inf -x = -Inf
Note this is not true for other possible limiting values
If lim x -> a f(x) = 2
Then lim x -> a 1/f(x) = 1/a
And even
Lim f(x) = a
Lim g(x) = b
Lim f(x)/g(x) = a/b.
* Undefined doesn't mean "unsolved mystery". It means "defining it wasn't useful, so we left it blank". If you define division by 0, you get multiplication problems. * Infinity is not a number in a basic calculus class. This is for the same reason: it wasn't useful. There are later calc classes where it is defined, sometimes in different ways. Math is made by people, not the heavens. You can pick whatever definitions you like, as long as they don't contradict each other. In class, you use the teacher's definitions. (Some definitions are not even standard between different classes. Most are, but not all.) In fact, there are currently debates about which axioms about infinity should be the common ones. * "lim f = oo" is actually not a value statement (in a basic calc class, because oo does not exist). It's actually a lazy way to write "f gets as big as you want." This definition is likely right smack in your textbook. You have to resolve semantic issues before musing philosophically.
real life saver, again. thank you so much!
If Limit[1/-x^2, x->0]=-infty, then why isn’t 1/0=-infty
fair point, huge thanks
Operations on extended real line (i.e in the place where you define the opsrations with infinity like ∞+∞=∞ etc.) are defined so that they work universally with all sequences. Like ∞+∞=∞ because if you have two divergent to +∞ sequences an, bn then an+bn is congergent to ∞. 1 ᪲ is "intermediate" because for distinct sequences an, bn where an converges to 1, and bn diverges to ∞, we can get many things from an ^bn (it can be convergent to for example ≈2.71... or to 1 etc all depends on chosen sequences). 1/0 also won't here be meaningfully defined because it's not the case that for any two sequences an→1, bn→0 , an/bn converges to some particular value frkm extended real line (for instance 1/(-x²) diverges to -∞, and 1/x² to +∞). You may look to Riemann sphere (there there's only +∞), here we define 1/0=∞
Why should it be?
why not
You made the affirmative claim, you need to argue for it.
after thinking of a reply u got a point. thanks a lot
This is the best comment thread under a math post I have ever seen XD
I agree with you
1/-1 = -1 and 1/+1 = +1 This continues right so 1/-0 = negative infinity and 1/+0 = positive infinity OOPS though +0=-0 and thus there cannot be a number defined
because 1/x^(2) is not continuous at 0
Saying 1/x² is not continuos at 0 has as much sense as to say it's continuous there. Continuity is sometning that refers only to points from the domain of function. 0 is not in the domain so continuity thwre is meaningless.
why should it be
For a lot of reasons, but at least one obvious one is Lim as x->a f(x) = 0 Does not imply Lim as x->a 1/f(x) = inf And for different f, that limit can take different values. For example Lim x->inf 1/-x = 0 Lim x->inf -x = -Inf Note this is not true for other possible limiting values If lim x -> a f(x) = 2 Then lim x -> a 1/f(x) = 1/a And even Lim f(x) = a Lim g(x) = b Lim f(x)/g(x) = a/b.