When deriving the fundamental solution in his book PDE (section 2.3.1, page 46), Evans comes to the equation $$r^{n-1}w'+\frac{1}{2}r^n w=a,$$ where $w:=w(y)=w(|y|)=w(r)$. After that he assumes that if $\lim_{r \rightarrow\infty}w, w'=0$, then $a=0$. But if so, we will have $\lim_{r \rightarrow\infty}(r^{n-1}w')=\infty\cdot0$ and $\lim_{r \rightarrow\infty}(r^{n}w)=\infty\cdot0$.
Applying the L'Hospital's Rule wouldn't help much since we don't know the behaviour of higher derivatives of $w$. My question is how to obtain the result $a=0$? I suppose it is pretty straight forward, but unfortunately can't see it.
Thanks everyone in advance.