자연계열 1번

$\displaystyle g(1)=2$에서 $\displaystyle c=1$$\displaystyle \lim_{x\to\infty}\frac{g^{\prime}(x)}{f(x)}=\lim_{x\to\infty}\frac{nx^{n-1}+lx^{l-1}}{ax^{m+1}+bx^m}=1$에서 $\displaystyle n=m+2,\;a=m+2$

$\displaystyle \lim_{x\to0}\frac{xg(x)}{f(x)}=\lim_{x\to0}\frac{x^{n+1}+x^{l+1}}{ax^{m+1}+bx^m}=\frac14$에서 $\displaystyle l=m-1,\;b=4$$\displaystyle f(1)=8$에서 $\displaystyle m=2$이고 $\displaystyle a=4,\;b=4$.따라서, $\displaystyle a=4,\;b=4,\;c=1$.

$\displaystyle f(x)=4x^3+4x^2$이고 $\displaystyle g(x)=x^4+x$이므로$\displaystyle \begin{aligned}L=\lim_{x\to-1}\frac{g(x)}{f(x)}&=\lim_{x\to-1}\frac{x^4+x}{4(x^3+x^2)}=\lim_{x\to-1}\frac{x^3+1}{4x(x+1)}\\&=\lim_{x\to-1}\frac{x^2-x+1}{4x}=-\frac34\end{aligned}$