$($For each given limit$,$ if the numerical value of the limit exists, enter the answer as an integer or as a simplified fraction in lowest terms; if the limit doesn't exist, enter "DNE" in uppercase.$)$

Let $f(x)=\frac{|x+1|(x+2)}{(x+1)(x+7)}.$

$($a$)$ Find $\underset{x-1^{+}}{lim}f(x)=?$

$($b$)$ Find $\underset{x-1^{-}}{lim}f(x)=?$

$($c$)$ Find $\underset{x-1}{lim}f(x)=?$