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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 7
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PROFESSOR: Right now, we're finishing up with the first unit, and I'd like to continue
in this lecture, lecture seven, with some final remarks about exponents. So what I'd
like to do is just review something that I did quickly last time, and make a few
philosophical remarks about it. I think that the steps involved were maybe a little
tricky, and so I'd like to go through it one more time. Remember, that we were
talking about this number ak, which is (1 1/k)^k. And what we showed was that
the limit as k goes to infinity of ak was e.
So the first thing that I'd like to do is just explain the proof a little bit more clearly
than I did last time with fewer symbols, or at least with this abbreviation of the
symbol here, to show you what we actually did. So I'll just remind you of what we
did last time, and the first observation was to check, rather than the limit of this
function, but to take the ln first. And this is typically what's done when you have an
exponential, when you have an exponent. And what we found was that the limit here
was 1 as k goes to infinity.
So last time, this is what we did. And I just wanted to be careful and show you
exactly what the next step is. If you exponentiate this fact; you take e to this power,
that's going to tend to e ^ 1, which is just e. And then, we just observe that this is
the same as ak. So the basic ingredient here is that e ^ ln a = a. That's because the
ln function is the inverse of the exponential function. Yes, question?
STUDENT: [INAUDIBLE]
PROFESSOR: So the question was, wouldn't the log of this be because ak is tending
to 1. But ak isn't tending to 1. Who said it was? If you take the logarithm, which is
what we did last time, logarithm of ak is indeed k(ln(1 1/k)). That does not tend to
0. This part of it tends to 0, and this part tends to infinity. And they balance each
other, times infinity. We don't really know yet from this expression, in fact we did
some cleverness with limits and derivatives, to figure out this limit. It was a very
subtle thing. It turned out to be 1. All right?
Now, the thing that I'd like to say - I'm sorry I'm going to erase this aside here - but