# Napier's Constant Limit Definition

• scorsesse
In summary, the conversation discusses the limit of the function (1+ 1/x)^x as x approaches infinity and its relationship to Napier's constant, e. Different methods for proving this limit exist, including defining e as the limit itself or through the derivative of the function a^x. Ultimately, the value of e is closely related to the expression (1+ 1/x)^x as x goes to infinity.
scorsesse
Hi all ! I am terribly sorry if this was answered before but i couldn't find the post. So that's the deal. We all know that while x→∞ (1+1/x)^x → e

But I am deeply telling myself that 1/x goes to 0 while x goes to infinity. 1+0 = 1 and we have 1^∞ which is undefined. But also see that 1/x +1 is not a continuous function so i cannot simply take the limit of it and raise the value to x like : (limit of 1/x + 1)^x while x→∞

So can you please give me a rigorous proof for why this function approaches to Napier's constant ?

You cannot first take the limit of 1+ 1/x as x goes to infinity and then say that you are taking $1^\infty$. The limits must be taken simultaneously.

How you show that $\lim_{x\to \infty}(1+ 1/x)^x= e$ depends upon exactly how you define e itself. In some texts, e is defined as that limit, after you have proved that the limit does, in fact, exist.

But you can also prove, without reference to e, that the derivative of the function $f(x)=a^x$ is a constant (depending on a) time $a^x$. And then define e to be such that that constant is 1.

That is, if $f(x)= a^x$ then $f(x+h)= a^{x+h}= a^xa^h$ so that
$$\frac{a^{x+h}- a^x}{h}= \frac{a^xa^h- a^x}{h}= a^x\frac{a^h- 1}{h}$$
so that
$$\frac{da^x}{dx}= a^x \lim_{h\to 0}\frac{a^h- 1}{h}$$
and e is defined to be the number such that
$$\lim_{h\to 0}\frac{a^h- 1}{h}= 1$$.

That means that, for h sufficiently close to 0, we can write
$$\frac{a^h- 1}{h}$$
is approximately 1 so that
$$a^h- 1$$
is approximately equal to h and then $a^h$ is approximately equal to 1+ h.
That, in turn, means that a is approximately equal to $(1+ h)^{1/h}$

Now, let x= 1/h so that becomes $(1+ 1/x)^x$ and as h goes to 0, h goes to infinity.

## 1. What is Napier's Constant?

Napier's Constant, denoted by the symbol e, is an irrational number approximately equal to 2.71828. It is also known as Euler's Number, named after the Swiss mathematician Leonhard Euler.

## 2. What is the limit definition of Napier's Constant?

The limit definition of Napier's Constant is the mathematical expression (1 + 1/n)^n as n approaches infinity. This expression is equivalent to e and can be used to approximate the value of e.

## 3. How is Napier's Constant related to exponential functions?

Napier's Constant is the base of the natural logarithm, which is the inverse of the exponential function. This means that e raised to any power will equal the input of the natural logarithm function.

## 4. What is the significance of Napier's Constant in mathematics?

Napier's Constant plays a crucial role in many areas of mathematics, including calculus, complex analysis, and number theory. It is also used in various scientific and financial applications, such as population growth models and compound interest calculations.

## 5. How is Napier's Constant calculated?

Napier's Constant cannot be calculated exactly, as it is an irrational number with infinite decimal places. It is typically approximated using the limit definition or by using a series expansion, such as the Taylor series for the exponential function.

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