Math Is Fun Forum

The limit of 1⁄x as x approaches 0 doesn't actually exist, since the answer is different depending on the direction from which x approaches 0.

We can say that the limit of 1/x approaches ∞ as x approaches 0 from the positive side, and that the limit of 1/x approaches -∞ as x approaches 0 from the negative side. We can also say that the "limit" of the absolute value of 1⁄x approaches ∞ as x approaches 0.

It's just a small detail, but an important one for those who are just learning the concept of limits.

I wasted a few hours yesterday trying to solve this puzzle.

x≈3.53208888624 or x≈2.34729635533 or x≈0.120614758428

I'm pretty sure that's not quite what you were looking for.

It's the shuffle, not the deal, that determines which cards we get. (Note that game rules typically dictate the order in which cards must be dealt from the deck.) Just because we don't see the value of a particular card until we see that card's face doesn't mean that that particular card did not have that same value prior to us seeing the face.

The only way we can have free will is if the information about our future is unknowable to every possible observer, including god. If that is the case, then god cannot be omniscient. Omniscience implies there is no unknowable information. If god is omniscient, then god's knowledge of our future implies that we have a particular path through life from which we are unable to deviate, even if we, personally, do not see that path. The cards are already shuffled, and we're just waiting for them to be dealt to us.

Our perception of free will does not imply free will. Our own lack of knowledge of the value of the next card does not mean the next card doesn't have a fixed value prior to us seeing it. The existence of an omniscient entity precludes free will. Such an entity would know the value of our next card. Free will implies that the future cannot be knowable. Free will implies that we are able to assign the value of our next card as we obtain it, and prior to that assignment of value, that card had no particular value, as opposed to having a particular value that was unknown to us. I don't think we have reason to believe we could tell the difference between the two scenarios from our own perspective.

If the information from our future exists, as would have to be the case with an omniscient creator, then that implies a lack of free will. We can't really decide things as we go if the knowledge of all of our future decisions is known.

What about infrared, ultraviolet, x-ray, etc.?

If we are only concerned with light visible to the human eye, don't we just need red, green, and blue (RGB)?

They work in very similar ways. Experimental scientists, however, typically don't have the luxury of substituting x, y, and z (i.e. arbitrary variables) in place of constants to demonstrate that something holds for all cases, so they have to do much more testing. Theoretical scientists rely solely on mathematics and imagination.

Agreed.

This only holds true with experimental science, not theoretical science. Relativity Theory, for example, was established long before its implications were ever observed.

Also, mathematics have relied on observation throughout their development also. The constant relationship between a circle's circumference and it's diameter was observed to be a constant long before anyone understood what or why it was so. The 3-4-5 right triangle was observed to be a right triangle before the Pythagorean theorem was derived.

They, too, have their limitations. For example, we all know that the inside angles of any triangle sum to 180º. Except that only holds true for Euclidian spaces, which only really exist in manmade constructs. It is a convenient tool, since we live in the midst of man made constructs, but it holds no absolute truth. Certain conditions must be assumed, such as parallel lines never intersecting.

Sure it can. All one needs to do is show that it does not hold true in non-Euclidean spaces, and that non-Euclidean spaces exist. At that point, the Pythagorean theorem has to be modified to specify that it only applies in Euclidian space.

Tests? As in mathematical experiments?

Proof by contradiction is how the experimental process is used in attempt to disprove hypotheses. It is the cornerstone of the scientific method. Science does not have the ability to prove things to be true, only the ability to prove things are not true. Nothing is known in science with 100% certainty, except those things that are defined (e.g. we know the speed of light with 100% certainty because we have defined the meter's length in terms of the distance light travels in a vacuum in a fixed amount of time). All that can be known is that a hypothesis has been rigorously tested many, many times, and has never been disproved.

Science and mathematics are tightly interwoven. Neither would exist without the other. They share many methodologies. In many ways, mathematics are science, and in many ways, science is a branch of mathematics.

MS software excepted, of course.

Option 1. We don't know which penny came up heads, only that at least one of the two did. The one of the two that did come up heads was minted in 1980.

I know that any answers are going to be in terms of P(a).

I'm not sure I follow how you arrived at your answer.

True. There are 11 other possible (approximate) values of x, although the exact answer I listed actually includes all of them.

An efficient design would have the "e" at either the current j or f position. The home keys should be where the most often used letters are located.

There are other layouts available that are designed more efficiently.

Early typewriter would jam if keys were pressed too quickly in succession, hence the need to slow down typists. This is not an issue with computers, but the same inefficient key layout still prevails.