3. In the Cartesian coordinate system $x O y$, points with both coordinates as integers are called integer points. It is known that $k$ is a real number. When the intersection point of the two different lines $y=4 k x-\frac{1}{k}$ and $y=\frac{1}{k} x+2$ is an integer point, the number of values that $k$ can take is ( ).
(A) 1
(B) 2
(C) 3
(D) More than 3
3. C.

When $x=0$, $y=2$, so
$$
2=-\frac{1}{k} \Rightarrow k=-\frac{1}{2} \text {. }
$$

At this point, the two lines coincide.
Therefore, $x \neq 0$, and $k \neq-\frac{1}{2}$.
Obviously, $k \neq 0$, then
$4 k x-\frac{1}{k}=\frac{1}{k} x+2$
$\Rightarrow k=\frac{x+1}{2 x}$
$\Rightarrow y=\frac{2}{x+1}+2 x$.
Since $x$ and $y$ are integers, we have
$$
x+1= \pm 1, \pm 2 \Rightarrow x=-2,-3,1 \text {. }
$$

Accordingly, $k=\frac{1}{4}, \frac{1}{3}, 1$.
Upon verification, $k=\frac{1}{4}, \frac{1}{3}, 1$ satisfy the conditions.