u/ElegantPoet3386

Edit: There's a typo in my title, replace still be continuous with still exist

Suppose we have f(x) = sqrt(x).

When taking the limit of f as x approaches 0, something interesting happens. The left hand limit doesn't exist since well, the function isn't even defined to the left of x = 0.

The right hand limit of course is 0.

The question now is, what's the limit? If we go by the standard method of checking if the left and right hand limits are equal and assignning the limit value to what the 2 sided limits are equal to if they're both equal, the limit wouldn't exist since DNE doesn't equal 0 (also that doesn't exactly make sense since DNE isn't a number).

Does the limit actually fail to exist though? I know the delta epislon defintion probably has the answer to this but I do not know how to use it well enough.

I’ve heard that you can look at their last spurt posture, but I Unfortunately am unable see any difference. It would be helpful if there was something specific in the posture that indicated theyre out of stamina

The obvious answer is, well yes.

However, looking at the math behind the temperatures of the 2 objects as a function of time, the temperatures seem to asymptotically approach a common temperature due to being modelled by exponential decay. This means math is saying that they should never be able to reach the same temperature.

I can't think of a reason why the temperatures of 2 objects in contact won't eventually be the same after a finite period of time though. So should I trust what the math is telling me in this case?

Suppose we have something like f(x) * dy/dx + f'(x)*y = g(x).

Since the left side is just the product rule, we can rewrite this the LHS as d/dx(f(x)*y) = g(x). I think.

If we then take the integral of both sides, we should(?) get f(x) * y = integral g(x) dx

And then y = integral g(x) dx / f(x).

This seems like a really niche form, but are we allowed to reverse derivative rules to help us solve differential equations? And if so, is there anything stopping us from doing the same thing to the chain rule and power rule?

*Note my calculus knowledge only goes up to Calc 1

So, on the final day of my math class, my professor gave us a few simple challenges to think about, and one of them was to try to prove why -(-a) equals a. The key here was prove, not explain, so I can't just say turn yourself around twice.

It got me thinking for a few days as how to do this because it seems so simple so I don't even know where to start. This is what I eventually settled on.

The - symbol in math seems to have 2 main uses. If used between 2 numbers, like a-b, it can be thought of as a + -b. But that brings into question: what does -b mean?

What I eventually settled on was we can think of -b as the number x that satifies the equation b + x = 0. This seems to be a defintion I can't find issues with, but I'm not entirely sure.

Now, what does -(-a) mean? By the previous defintion, -(-a) would be the number x that satifies the equation x + -a = 0. Now, since -a is intended to be the number we add to a to get 0, it should also follow that if we add a to -a, we get 0. Thus, it should follow x is a.

Therefore, -(-a) equals a.

I'm not exactly sure if this counts as a proof, but I had fun thinking about how to prove it so even if this turns out to have holes, I'm happy. If real analysis is like this, I can't wait to take it in like 4 or so years.

For example, theroums like the Fundemental Theroum of Algebra. They tell you that this polynomial has say 4 complex roots. That's nice to know, except I don't actually know where those roots are. Or something like the Mean Value Theroum. It's nice to know there's a c value in (a,b) such that f'(c) = f(b)-f(a) / b - a, but it doesn't tell you where that c is.

I mean to me these kind of theroums seem cool, but they don't seem to exactly help you. However they have to be important otherwise things like the Fundemental Theroum of Algebra wouldn't be called fundemental and would otherwise just be named after the person who discovered it.