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### Course: Finance and capital markets>Unit 1

Lesson 4: Continuous compound interest and e

# 𝑒 as a limit

Sal continues the discussion on e, this time digging deeper into the mathematical definition of 𝑒. Created by Sal Khan.

## Want to join the conversation?

• so if the interest was lets say 7% instead of 100% would you put .07/n or instead of 1/n?
• Exactly! The general compound interest formula is (1 + r/n)^n, where r is the rate. Obviously 100% = 1 and 7% = 0.07, so you did a good job.
• What other applications are there for e?
• also if you ever see ln or ln() that is a reference to e as the base for log. Log base e of a number is called the natural log
• I don't understand what limits are when Sal talks about it at . Can someone explain?
• Let's consider the function:
𝑓(𝑥) = [1 + (1/𝑥)]ˣ
If we plug in larger and larger values of 𝑥, we see that the value of 𝑓(𝑥) seems to be getting closer and closer to a certain value around 2.71.... It turns out, when we use an infinitely large value for 𝑥, we get the exact value of 𝑒. More succinctly, we can say that the limit of 𝑓(𝑥) as 𝑥 tends to ∞ is 𝑒. Essentially, the limit helps us find the value of a function 𝑓(𝑥) as 𝑥 gets closer and closer to some value. You will learn more about limits and a more rigorous definition later in Precalculus and Calculus.
• Will this work if n is equal to a very large number i.e 999,999,999,999,999 because i tested it and the results were more than 3 so is there like a limit for n?
• Interestingly Google's "hidden" calculator does the same thing. I'm pretty sure this is a result of the limited precision with which floating point numbers are stored.

In other words, since computer chips can't store an infinite number of digits (that would take an infinite amount of of memory) numbers for intermediate steps get approximated and this can lead to inaccuracies like the one you noticed.

So, yes this works better for larger numbers, but only if you keep all of the information!

http://stackoverflow.com/questions/2100490/floating-point-inaccuracy-examples
• Sal references another video or videos at . What are those videos?
• It might be covered in some video later in this sequence of logarithm-related videos, though I doubt it, but it is mentioned in Vi Hart's pi videos under Recreational Math -> Doodling in math class.
• Instead of using 100% in (1+100%/n)^n can we take any other interest rates? Will that reach e as well?
• If you use a number other than 100% for the infinite series you won't get `e`, no.
• are there any other magical numbers other than pi e and i
• does this mean that (1+1/∞)^∞=e?
• 1. ∞ is merely a concept, rather than something you can substitute into an equation.

2. lim x->∞ means x converges to ∞, but never reach ∞. x can reach 100, can reach 10^10, can also reach 10^10000000, but never reach ∞.
• I can't make an intuitive sense of why it happens. As x approaches infinity, 1/x should approach 0(since, 1/infinity = 0). (1+0)^infinity should be 1 because 1 raised to any power is 1. So, shouldn't (1+1/x)^x be 1 as x approaches infinity? Why 2.7182818...?
• You could just as well say that (1+1/x) is greater than 1, and (something greater than 1)^x goes to infinity as x goes to infinity, so the limit should go to infinity.

The answer is that you don't get to pick parts of the limit to evaluate first. The exponent is going to infinity and 1/x is going to 0 at the same time. When you evaluate the limit your way, you're assuming the exponent is constant while 1/x goes to 0, which it isn't.
• So what actually IS e? He says what pi represents but never explains what e represents. All he said was that is was some cool number. plz help