h goes to 0 of f(h)- f(0) over h.

If we do this limit from the right, then we can evaluate f(h) as

e to the -1 over h.

Now, f(0) is, of course, 0.

And so we're left with the limit as h goes to 0 from the right

of e to the -1 over h divided by h.

If we change variables and let t be 1 over h, then this is the same thing

as the limit as t goes to to infinity of t times e to the -t.

Now that we know because exponential beats polynomial.

This is 0.

So I claim that with a little bit more work using a similar approach,

you can show that all of the derivatives of f at 0 are exactly 0.

That means that if you take the Taylor expansion of this function,

it exists, and it is precisely 0.

Every single coefficient in the Taylor series is 0.

However, the function itself is positive for

values of x that are strictly bigger than 0.

This is a smooth function, but it not is real-analytic.

This leads us to an image of the universe of functions,

beginning with the simplest functions.

At the very core of this universe lie the polynomials.

These are themselves divided or graded into different realms,

beginning with the constants and then the first order polynomials,

the quadratics, the cubics, etc.

Each filling out, degree by degree, larger and

larger subspaces of simple functions.

But this is not all there is,

since polynomials have power series that eventually terminate.

Beyond the space of polynomials lie those functions whose

Taylor series exist and converge to the functions.

That is the real-analytic functions, like e to the x, sine of x, cosine of x,

all those beautiful functions we've been working with all term.

However, beyond these still lie

other functions, more mysterious functions for

which Taylor expansion is not sufficient to describe them.

One normally doesn't run into such things.

That's why we haven't focused too much attention on them.

But you should know that they lie out beyond the real-analytics.

This leads us to our last image of what Taylor expansion really is.

Taylor expansion can be thought of as projection to

the space of polynomials, where computing a Taylor

polynomial is a projection to one of these finite subspaces.

Throwing away the higher ordered terms really is a form of projection.