Adição de little-o's

Let's review: which of these are in the class $o(x^2)$ as $x \rightarrow 0$ ?

Now suppose we have two functions which are both $o(x^2)$ , say $x^3$ and $x^2 \sin x$ .

Which of these is an expression for their sum?

Does $x + \sin x$ tend to zero as $x$ tends to $0$ ?

So... is $x^3 + x^2 \sin x$ in the class $o(x^2)$ as $x \rightarrow 0$ ?

Main result: If two functions are both $o(x^2)$ , then so is their sum!

Let's walk through why this is true.

If a function $f$ is $o(x^2)$ , then...

So, let's take another function $g$ which is also $o(x^2)$ , which means that there's some $G$ tending to $0$ , where $g$ is $x^2$ times $G$ .

The main observation is that:

Therefore, the sum $f+g$ is precisely $x^2$ multiplied by something that tends to $0$ , and so when $f$ and $g$ are both $o(x^2)$ , then $f+g$ is $o(x^2)$ too! We've finished the proof.

What about differences? What about $f-g$ ?

That's all, folks!

Adição de little-o's

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