Extension: Adding with Little-o Notation

$\frac{1}{2}x^2$

$x^3$

$x$

$x^2 \cos x$

$x^2 \sin x$

$x^2 (x + \sin x)$

$x^3 (1 + \frac{\sin x}{x})$

$x^5 \sin x$

$x^3 + x^2 + x^2 ( \sin x - 1 )$

Yes

No

Yes

No

It can be written as $x^2$ times some other function $F$, where $F$ tends to $0$.

It is less than $x^2$ for all $x$

$F + G$ tends to $0$ too.

Pigs can fly.

We could just do the same proof again, and use the fact that $F-G$ also tends to $0$.

The same proof won't work, because $F-G$ doesn't necessarily tend to zero.

When $g$ is $o(x^2)$, then so is $-g$, so this follows from the sum result for $f + (-g)$.