3. Let the equation of the ellipse be $\frac{x^{2}}{3}+y^{2}=1$, and $A(0,-1)$ be one endpoint of the minor axis, $M$ and $N$ be two distinct points on the ellipse. If there always exists an isosceles $\triangle A M N$ with $M N$ as the base, then the range of the slope $k$ of the line $M N$ is ( ).
(A) $(-1,0]$
(B) $[0,1]$
(C) $(-1,1)$
(D) $[-1,1]$
3. (C).

Let $M\left(x_{1}, y_{1}\right)$ and $N\left(x_{2}, y_{2}\right)$, it is easy to know that $x_{1} \neq x_{2}$. Suppose the equation of $MN$ is $y=k x+b$. Substituting into the ellipse equation $x^{2}+3 y^{2}=3$ yields
$$
\left(1+3 k^{2}\right) x^{2}+6 k b x+3 b^{2}-3=0 \text {. }
$$

Its two roots are $x_{1}$ and $x_{2}$.
From $|A M|=|A N|$ we get
$$
x_{1}^{2}+\left(y_{1}+1\right)^{2}=x_{2}^{2}+\left(y_{2}+1\right)^{2} \text {, }
$$

then $\left(x_{1}-x_{2}\right)\left(x_{1}+x_{2}\right)+\left(y_{1}-y_{2}\right)\left(y_{1}+y_{2}+2\right)=0$.
$$
\begin{array}{l}
\text { Also, } \because y_{1}=k x_{1}+b, y_{2}=k x_{2}+b, \\
\therefore y_{1}-y_{2}=k\left(x_{1}-x_{2}\right), \\
y_{1}+y_{2}+2=k\left(x_{1}+x_{2}\right)+2 b+2 .
\end{array}
$$

Substituting into equation (2) gives $\left(1+k^{2}\right)\left(x_{1}+x_{2}\right)+2 k(b+1)=0$.
Substituting $x_{1}+x_{2}=\frac{-6 k b}{1+3 k^{2}}$ into the above equation yields $2 b=1+3 k^{2}$.
Furthermore, from the quadratic equation (1) having two distinct real roots, we have
$$
\Delta>0 \Rightarrow b^{2}<1+3 k^{2} \Rightarrow k^{2}<1 \Rightarrow-1<k<1 .
$$