Proof: e^x is Continuous using Epsilon Delta Definition | Real Analysis Exercises

[@awvz_1194]

If you are using the power series representation for e^x, you are assuming e^x is analytic, which implies that e^x is continuous, which is circular logic

[@3141minecraft]

No, it is not. Because you can define e^x as that power series, and you can prove e^(x+y)=e^x*e^y with that definition. It might be hard to prove it is positive for all real numbers, which is used at 3:35 to remove the absolute value, but you can just not remove the absolute value and proof will still work.

[@mzg147]

Here we don't know that analytic implies continuous, in fact this generalization you are talking about is similar to prove

[@kingbeauregard]

Questions: 1) As a general rule, is it logically sound to do Taylor series to find a relationship between delta and epsilon? My concern is if the Taylor series somehow assumes the existence of a limit -- in which case our proof is circular -- but I guess that's not actually a problem, right? 2) Being very formal, delta needs to be the minimum of 1 and epsilon / (epsilon + 1), right? 3) In most of the examples I've seen, the relationship between delta and epsilon is linear; it's not in this case. Do I take it that really all that's required is, as delta goes to zero, so does epsilon?

[@kingbeauregard]

Wait, I can answer my #2: since epsilon / (epsilon + 1) is always less than or equal to 1, we don't need to do the minimum. About #3, I have since decided that linear relationships are typically used as examples because they're easier to explain, but there's nothing about epsilon-delta proofs that requires it. Although, I have to say, I feel it's a lot easier to prove the limit of e^x at x = 0 if I take what I know about e^x, and set delta = min { 1, epsilon / (e - 1) }. I get that by drawing the line that goes through (0, 1) and (1, e), and observing that e^x is greater than the line for x < 0, and is less than the line for 0 < x

[@Bedoroski]

Exactly what we needed ❤

[@WrathofMath]

Thanks for watching!

[@Kane88168]

how about e^(-x)? are the steps still the same, just that we change the x into -x ?

[@krasimirronkov17]

Can you make a playlist on polynomials in depth

[@WrathofMath]

Thanks for watching, and we'll certainly cover lots about polynomials in various ways in playlists I am currently working on. What is it you want to know about polynomials?

[@krasimirronkov17]

@@WrathofMath The thing I'm struggling with is the theorem for finding the upper and lower bounds on the roots of a polynomial. There are some formulas that I dont understand exactly

[@WrathofMath]

Which formulas do you mean? I haven't looked at that sort of thing in a while so i don't know what it is off the top of my head. Do you mean the kind of thing described here? www.mathsisfun.com/algebra/polynomials-bounds-zeros.html

[@odobenusrosmarus6035]

so do we not need to fill in that first delta?

[@WrathofMath]

Thanks for watching, Super Awesome Guy! Super awesome question - my proof videos are general outlines/explanations of how the proof would go, so I usually don't give a fully written proof in the video, but a fully detailed explanation of the points a proof would cover. So in this case, if we wanted to rework that into a final proof, we would prove that e^x is continuous at 0 first. Then we would begin our proof that e^x is continuous at all real numbers. We'd take epsilon>0 and take c in R, then we might point out that |f(x)-f(c)| = |e^x - e^c| = e^c|e^{x-c} - e^0|. Letting y=x-c we know e^y is just the exponential function (which we proved is continuous at 0), since it has the same domain and same image for each input. So we know there exists delta>0 such that for all y in R with |y-0|=|y|

[@odobenusrosmarus6035]

@@WrathofMath I see, thank you!! Love your videos btw!

[@Leo-io4bq]

🤙🤙

[@_.lilkay_2577]

Soft

[@aashsyed1277]

Hey it's me

[@WrathofMath]

Hey aash, good to see ya!

[@aashsyed1277]

@@WrathofMath I want to tell you that I will create a youtube channel sometime like 3 years from now and by the way I am 9 years old but I can understand CALCULUS and I live in pakistan