Given that limit exists so left hand limit must be equal to the right hand limit.
$\displaystyle \lim_{x \to -1^-}f(x) = \lim_{x \to -1^+}f(x)$
$\implies \displaystyle \lim_{x \to -1^-} x+2 = \displaystyle \lim_{x \to -1^+}cx^2$
$\implies -1+2 = c$
$\implies c = 1.$
So, correct option is A.