A problem on power series from chapter 24 of Spivak Calculus 3rd Edition

Question

Please refer part (b) in the enclosed image.
$f(x)=e^{-1/x^2}\sin (1/x), x\ne 0$ cannot have a power series about $x=0$ because we can take the sequence $x_n=(1/n\pi)$ and noting that $f(x_n)=0$, we conclude from part (b) that all coefficients of the power series are $0$.
However I don't understand the yellow highlighted portion in the enclosed image. Because if it is true then consider $f(x) =\sin(x+a^2)=\sum_{j=0}^{\infty}(-1)^j \frac{(x+a^2)^{2j+1}}{(2j+1)!}$ for some real $a$. It is clearly $0$ at $x=-a^2$.
What does the yellow highlighted portion want to say? Can you please help me understand it by giving examples if possible? Thanks in advance.

Answer 1

It is talking about a function that is $0$ for all $x \le 0$ (or at least all $x$ in some interval $(-\varepsilon, 0]$).

Answer 2

Clearly $a_0=f(0)=0$. Now for $x\neq0$ in $(-R,R)$, $f(x)=0$ and hence $$ \frac{f(x)-f(0)}{x}=0. $$ Letting $x\to0$, one has $$ f'(0)=0$$ which implies $a_1=0$. Repeating this, one concludes $a_n=0$ for $\forall n$.