Answer
$\lim\limits_{x\to1}\ln\Big(\frac{5-x^2}{1+x}\Big)=\ln2$
Work Step by Step
$$A=\lim\limits_{x\to1}\ln\Big(\frac{5-x^2}{1+x}\Big)$$
Let $f(x)=\ln x$ and $g(x)=\frac{5-x^2}{1+x}$, we have $$A=\lim\limits_{x\to1}f(g(x))$$
Since $\ln x$ is continuous on $(0,\infty)$, we can apply Theorem 8: $$A=f(\lim\limits_{x\to1}g(x))$$
$g(x)$ is a rational function, so it is continuous on $R$ except where $x=-1$.
Therefore, $\lim\limits_{x\to1}g(x)=g(1)$
$A=f(g(1))$
$A=\ln\Big(\frac{5-1^2}{1+1}\Big)$
$A=\ln2$
