Calculus! #40: Early Transcendentals 3.6B

The Number e as a limit.

I want to lead off with the way I like to explain e, then we’ll proceed to e as a limit, which is the more common way to think of it. The following isn’t a textbook definition and isn’t very rigorous. It’s mainly designed to give you a feel for e, and to remove any sense you may have of it as some magical number. It’s a special number, but it’s not magic.

Imagine it’s the 18th century and you’re on a boat somewhere and you need to make some nautical calculations. This involves taking the derivative of exponential functions. You know that the derivative of an exponential function is proportional to itself (i.e. the derivative of b^x = a*b^x, where a is some constant), but that doesn’t help you since the proportionality constant depends on the base of the function.

So, (2^x)' has a different proportionality constant than (3^x)', which has a different proportionality constant from (4^x)', and so on. Because you don’t have a calculator, you have a gigantic book that lists the proportionality constant for every nember from 0 to 100 to 3 decimal places. So, whenever you want to calculate the derivative (i.e. find the proportionality constant), you look up the number in the book. It’s a biiiig pain in the ass.

Then one day, you notice that the proportionality constants increase as the base increases. That is, 3 has a higher constant than 2, and 4 has a higher constant than 3, and so on. It occurs to you that there must be SOME number you can use for a base that has a proportionality constant of 1. That is, there is some base where shit gets really easy because ({base}^x)' = {base}^x.

You flip through the book and quickly note that 2′s proportionality constant is less than 1, and 3′s proportionality constant is higher than 1. So, this beautiful wonderful will-allow-you-to-be-way-lazier constant must be somewhere in between. After going through the giant book for a while, you finally arrive at a number around 2.718, which is extremely close to producing a proportionality constant of 1. (According to wolfram it’s good to 4 decimal places:

You decide to call this base “e” because everything’s freakin’ easy now, and you rejigger your system so all the exponential functions are base e. That way, once you know what function you’re dealing with, you can easily do a mental calculation, rather than having to use the giant book. You use all your spare time to get a buxom wench in every port, and live happily ever after until you die of neurosyphilis, raving to yourself on a deserted island somewhere near the Cape of Good Hope.

That’s how I like to think about e. HOWEVER, there is a much more succinct and mathematically beautiful (i.e. doesn’t involve syphilis) version. This is the most common definition of e, and has a really cute proof to boot.

SO, let’s dig in.

This one would be a real bitch in latex, so I’m just going to write out all the steps.

1) Because we define it that way

2) Proved in last section

3) Simple substitution of 1 into the equation in (2)

4) Definition of a limit

5) Fuck. This step is really confusing. We just stipulated x=1, and now we’re taking the limit as x goes to 0. How does that make sense?

I actually stopped here and looked for an instance of this step in any other proof of e as a limit. I failed to find it, then went to twitter asking if I was crazy.

It turns out the conclusion I came to (if my twitter geeks are right) was correct – we’re dealing with two different fucking versions of x. There’s the dependent variable version we stuck in f(x), and there’s the limiting variable version in the limit. Because the author is a sadist (according to one twitterer), he assigned two similar but distinct ideas to the same variable. That is, the x at the top is the same as the x at the bottom, but NOT the same as the x in the middle. Balls!

Fortunately, it turns out you can skip (5) and just leave everything in h form until the end. THEN, you can change the name of the variable to x or z or a drawing of a wiener for all I care.

So, I’m going to continue with the book’s proof because I am, after all, blogging this textbook. But, here I’m showing a quick proof that you can maintain the h the whole time, then after you’ve done the proof, you can change the variable to whatever you feel like.

Now, back to the proof.

6) Follows from (1)

7) Because ln(1) is 0

8) Log rule for powers says this is a legal move

9) Follows from (3)

10) Raise both sides as exponents to base e

11) This one’s a liiiiittle trickier. Look back to 125 for the theorem that states essentially this: If is continuous everywhere we care about, then the limit of f of g of x is the same as f of the limit of g of x. Read that 14 times and you’ll have it. Intuitively, I like to think of it this way: Either way you do it, you’re zooming in on a particular point. The first way, you figure out f(g(x)) then take its limit somewhere. The second way, you take the limit of g(x) somewhere, which gives you a value which you then run through f. Hopefully that’s not too confusing. If you mull it over for a bit, I think you’ll get it.

In practice, it means when you have a continuous nested function, you can pull the limit sign outside of the outermost function. And, that’s what we do in step 11.

12) The base e kills the ln, which simplifies things quite a bit.

13) Because e^1 = e.

14) They actually run the operation for smaller and smaller x and find that it’s somewhere around 2.7182818…




Okay, so that was a little painful, but the end result is this perdy little equation that allows you to easily calculate lots of digits of e.

Additionally, at the end, we get what I think of as the “classic” definition of e:

e=\lim_{n \to \infty} (1+\dfrac{1}{n})^n

All they did was say n = 1/x. This results in a version of the equation where you make the variable bigger and bigger, rather than closer and closer to 0.

Next stop: Rates of Change in the Natural and Social Sciences.



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6 Responses to Calculus! #40: Early Transcendentals 3.6B

  1. spyked says:

    A fact worth mentioning is that even though the above definition uses real functions like the natural logarithm, the limit definition of e doesn’t mention anything about real numbers. That is, the variable of the given sequence is a natural number.

    Sure, the other (more general, i’d say) definition of e as a case of Euler’s formula uses complex numbers, but this combination of natural numbers and limits to define a fundamental constant is one of the best examples of how calculus messes around with maths.

  2. Andrew says:

    So how do you prove that the antiderivative of ln(x) is a logarithm?

    It looks like you’re starting from the definition of ln(x) as the antiderivative of 1/x, which is fine. But then you have to assume, it seems, that ‘the antiderivative of 1/x’ is of the form log_b(x) for some b, which you will then figure out.

    • Andrew says:

      Sorry, that should have read:

      So how do you prove that ln(x) is a logarithm?

      It looks like you’re starting from the definition of ln(x) as the antiderivative of 1/x, which is fine. But then you have to assume, it seems, that ‘the antiderivative of 1/x’ is of the form log_b(x) for some b, which you will then figure out.

  3. Talithin says:

    It’s true that it’s a circular proof and so not strictly a proof. But I think for beginners, it at least gives some kind of intuition as to why this is a natural choice for the definition of e.

    I’ll be honest though, my favourite definition of e is its power series definition given by evaluating the Taylor polynomial of exp(x) at x=1. As a definition, it’s much more useful; as an intuitive concept, maybe not so much.

  4. Charlie says:

    Just a LaTeX note, if you use “\left(” and “\right)” for your parenthesis (or brackets or whatever), LaTeX will automatically size them correctly for you, which looks much better when you have fractions inside them. But you always have to use them in pairs!

  5. James says:

    Worth noting is that the “classical” definition has a really nice interpretation from the standpoint of banking, of all freaking things.

    If you had a bank account that got a fixed amount of interest (x) with more frequent payouts of interest to compensate for lower interest rates. I.e., paying out every x-th part of a time period, so that if we interpret one time period as a year and let x=0.25, you’re getting 25% interest on your account every 3 months. (Hey, I didn’t say it was a REALISTIC bank account)

    You could find your new balance after a year by calculating an accrued interest over that year and multiplying by the original balance, but the interest is what really matters. Written in math-y terms,

    I(x) = (1+x)^(1/x)

    If we let x->0, this means you’re getting almost NO interest, but you’re dividing the interest period into infinitesimal chunks of time. You’re getting interest every single moment, and every single subdivision of that moment. Even though the interest is craplousy for that small period, you’re getting paid out way more often than other accounts, so in the long run, you’re going to be getting megabucks (the exponential is going to grow faster than any polynomial of finite degree).

    So, one can say that e is tied up in this bizarre process of continuous interest, which seems ridiculous from a financial perspective, but plays very nicely into the processes that actually behave exponentially in nature. Radioactive matter, for example, doesn’t stop every N minutes and say “whoa whoa, we’re past the half-life, some of you pricks are going to have to move,” it just continually adjusts itself.


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