Can anyone please explain me this highlighted line-
My intital thoughts are as follows. If $f$ and $g$ have a common zero say $z_0$ then, if $$lim_{z\to z_0}(z-z_0)\frac{f(z)}{g(z)}=lim_{z\to z_0}(z-z_0)\frac{(z-z_0)^mf_1(z)}{(z-z_0)^ng_1(z)}=\frac{f_1(z_0)}{g_1(z_0)}lim_{z\to z_0}(z-z_0)^{1+m-n}=0$$ we will have a removal singularity and $1+m-n>0$. If $1+m-n\leq 0$ then $m<m+1\leq n$ and hence $$lim_{z\to z_0}\frac{f(z)}{g(z)}=\frac{f_1(z_0)}{g_1(z_0)}lim_{z\to z_0}(z-z_0)^{m-n}=\infty$$ i.e., a pole.
Am I correct? Also without using the fact that any meromorphic function is a quotient of two analytic functions can we prove that quotient of two meromorphic function is meromorphic?
Thanks in advance for any suggestion.