<s>[INST] Is there a mistake in solving this limit? I want to solve this:
\begin{equation}
L=\lim_{x\rightarrow 0}  \frac{\sum\limits_{m=1}^{M}a_{m}\exp\left\{\frac{bm^2}{ (m^2+2x^2)x^2}\right\}{(m^2+2x^2)^{-3/2}} }   {\sum\limits_{m=1}^{M} c_{m}\exp\left\{\frac{bm^2}{ (m^2+2x^2)x^2}\right\}{(m^2+2x^2)^{-3/2}}}
\end{equation}
where
\begin{equation}
\begin{aligned}
m=1,2,\cdots,M\\
a_m,b,c_m \neq 0 \quad \text{and are all constants}
\end{aligned}
\end{equation}
Here is my approach:
Since $x\rightarrow 0$, I ignore the term $2x^4$ in the denominator of $\exp\left\{\frac{b_m}{ (m^2+2x^2)x^2}\right\}$ and I get:
\begin{equation}
\begin{aligned}
L&=\lim_{x\rightarrow 0}  \frac{\exp\left\{\frac{b}{x^2}\right\}\sum\limits_{m=1}^{M}a_{m}{(m^2+2x^2)^{-3/2}} }   {\exp\left\{\frac{b}{x^2}\right\}\sum\limits_{m=1}^{M} c_{m}{(m^2+2x^2)^{-3/2}}}\\
&=\lim_{x\rightarrow 0}  \frac{\sum\limits_{m=1}^{M}a_{m}{(m^2+2x^2)^{-3/2}} }   {\sum\limits_{m=1}^{M} c_{m}{(m^2+2x^2)^{-3/2}}}\\
&=\frac{\sum\limits_{m=1}^{M}\frac{a_{m}}{m^3} }   {\sum\limits_{m=1}^{M} \frac{c_{m}}{m^3}}
\end{aligned}
\end{equation}
My question is:

*

*Is there a mistake in my derivation?


*If I made mistakes in the derivation, then, what is the correct derivation?
Thanks for helpful comments and answers!
 [/INST] Since @justt gave the answer, just try.
Take an example : $a_m=m$, $c_m=m^2$, $b=1$, $M=4$.
Without any simplification at all, the limit is
$$\frac{144+36 e^{3/2}+16 e^{16/9}+9 e^{15/8}}{144+72 e^{3/2}+48 e^{16/9}+36
   e^{15/8}}=0.465476$$
Ignoring the term $2x^4$, the limit becomes $\frac{41}{60}$
</s>