So we threw down a challenge at the end of our discussion of Tangent Lines, lets solve it.

Recall we wanted to **find a curve** such that for any we have the line tangent to that point satisfy .

(Historical note: Debeaune asked Descartes this problem in a letter in 1638; Leibniz solved this problem in 1684 using “infinitesimals”. We will use modern calculus to solve the problem!)

So what is ? Spoiler below!

Well, we set up the equation describing the tangent line

(1)

We can rewrite this to be

(2)

Or equivalently for any .

Lets try using the tangent line as an approximation for this curve. So for small we write

(3)

But since we rewrite the right hand side as

(3′)

What to do?

**Assumption:** Lets write .

We can have a coarse approximation

So for small we have

We could also break this down into two steps:

(4).

Why can we say this? Well, because

and plugging this into (4) gives the correct relation.

We could get a better approximation iterating this three times:

(4a).

We could do this times:

(4b).

Taking the limit we obtain the definition

(5).

For sufficiently large , we have for any the quantity be really small. So Equation (5) works for any real number.

What is this ? It is precisely the exponential function, usually written .

**Exercise 1.** Prove .

**Exercise 2.** What if we drop our assumption and allow be some nonzero real number? What if we let it be zero?

**Exercise 3.** Prove for any positive real number .

**Exercise 4.** Calculate to 7 digits of precision. (This mathematical constant “” is called “Euler’s constant”.) [Hint: we need for digits of precision to evaluate the limit with some .]

**Exercise 5.** Lets prove some inequalities involving ! Prove the following:

(a)

(b) for

(c) for

(d) for for

(f) for and (i.e., is a positive integer)

(g) for ,

(h) for .

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