Why Is lim g(t)/t = 0 as t -> 0?

[kingwinner]

"Little o" function

Claim: e^t= 1 + t +o(t)

Proof:
e^t = 1 + t + t²/2! + t³/3! +...
Let g(t)=t²/2! + t³/3! +...
g(t) is o(t), thus e^t= 1 + t +o(t).
=================

I don't understand why g(t) is o(t).
Why is it true that
lim g(t)/t = 0 ?
t->0

I know that for example lim[f(x)+g(x)]=lim f(x) + lim g(x), so if lim f(x)=0 and lim g(x)=0, then
lim[f(x)+g(x)] = 0+0 = 0. I believe this property is true only when computing the limit of a sum of a FINITE number of functions.

But g(t)/t above is an infinite sum; it has an infinite number of terms. How can we compute and prove that the limit of g(t)/t is 0? I know each term goes to 0, but we are summing an INFINITE number of terms, so how can you be sure that the limit is 0?

Any help/explanations would be much apprecaited!

 

[Char. Limit]

Well, the thing about our "infinite function" is that it can be rewritten in closed form like so:

$$g(t) = \sum_{n=2}^\infty \frac{t^n}{n!}$$

Now if this is g(t), then g(t)/t is like so:

$$\frac{g(t)}{t} = \frac{1}{t} \sum_{n=2}^\infty \frac{t^n}{n!} = \sum_{n=2}^\infty \frac{t^{n-1}}{n!}$$

Now it's easy to see that for all n>1, t^(n-1)/n! tends to zero as t tends to zero. So therefore, every term in the sum tends to zero, and the sum itself thus tends to zero.

 

[kingwinner]

So therefore, every term in the sum tends to zero, and the sum itself thus tends to zero.

I am having some trouble with this. If the upper limit of the sum is, say, 10, (i.e. a finite series where we are summing up a FINITE number of functions) then I can understand that the limit of the sum would be 0. There is a theorem/property of limits that guarantees this, which is lim[f(x)+g(x)]=lim f(x) + lim g(x) when the limits of the RHS both exist.
so if lim f(x)=0 and lim g(x)=0, then lim[f(x)+g(x)] = 0+0 = 0.

But for g(t)/t here, we are summing up an INFINITE number of terms that tend to 0, how can we justify or even rigorously prove that
lim g(t)/t = 0 ?
t->0

Thanks for the help!

 

[disregardthat]

So therefore, every term in the sum tends to zero, and the sum itself thus tends to zero.

That's not necessarily true for series not converging absolutely, I believe.

 

[Char. Limit]

That's not necessarily true for series not converging absolutely, I believe.

Then prove that it converges absolutely. I'm almost certain that this particular series does.

 

[disregardthat]

Then prove that it converges absolutely. I'm almost certain that this particular series does.

Surely this series does, since the terms are always of the same sign for positive t (which proves it for negative t as well). I was pointing out that the general statement does not hold.

 

[kingwinner]

OK! Now I have another question.

Definition: A function f is called o(h) if
lim [f(h)/h] = 0.
h->0

I understand what o(h) means, but in one of the examples from my textbook an o(√h) pops up, but I don't understand the meaning of o(√h).

Do we say f is o(√h) if
lim [f(h)/√h] = 0?
√h->0

or do we say f is o(√h) if
lim [f(√h)/√h] = 0?
√h->0

Thanks for explaining!

 

[disregardthat]

The first alternative, but as h --> 0.