= Solution
With the prescribed magnetic-only <Lorentz force> $m\ddot{\mathbf r}=q\dot{\mathbf r}\times\mathbf B$, the component equations are
$$
\boxed{\ddot x=\frac qm B_z(t)\dot y,\qquad
\ddot y=-\frac qm B_z(t)\dot x,\qquad
\ddot z=0.}
$$
Differentiate the squared planar speed:
$$
\frac d{dt}(\dot x^2+\dot y^2)
=2\dot x\ddot x+2\dot y\ddot y=0.
$$
Thus $V=\sqrt{\dot x^2+\dot y^2}$ is constant, since the <Lorentz force> is perpendicular to the velocity.
For $V>0$, introduce the velocity angle $\theta$ by $\dot x=V\cos\theta$, $\dot y=V\sin\theta$. The two planar equations imply $\dot\theta=-qB_z(t)/m$. Therefore the <velocity rotation in a time-dependent axial magnetic field> has phase
$$
\theta(t)=\phi-\frac qm\int_0^t B_z(s)\,ds.
$$
Equivalently, the complex velocity satisfies $\frac d{dt}(\dot x+i\dot y)=-iqB_z(t)(\dot x+i\dot y)/m$, giving the same integrated phase. Integrating the velocities and $\ddot z=0$ proves
$$
\boxed{\begin{aligned}
x(t)&=x_0+V\int_0^t\cos\left(\phi-\frac qm\int_0^{t'}B_z(s)\,ds\right)\,dt',\\
y(t)&=y_0+V\int_0^t\sin\left(\phi-\frac qm\int_0^{t'}B_z(s)\,ds\right)\,dt',\\
z(t)&=z_0+v_z t.
\end{aligned}}
$$
The constants $x_0,y_0,z_0$ are the initial position coordinates, $v_z$ is the constant velocity parallel to the field, $V\geq0$ is the initial planar speed, and $\phi$ is the initial planar-velocity direction modulo $2\pi$. Together they specify the initial position and velocity. If $V=0$, the planar position stays fixed and the angle $\phi$ is immaterial.
For $B_z(t)=\beta t$, the inner integral gives $\theta(t)=-at^2+\phi$, with $a=q\beta/(2m)>0$. Under the specified initial data,
$$
x(t)=\int_0^t\cos(as^2)\,ds,\qquad
y(t)=-\int_0^t\sin(as^2)\,ds,\qquad z(t)=0.
$$
Rescale $u=\sqrt a\,s$. The given <Fresnel integrals> then give
$$
x(\infty)=\frac1{\sqrt a}\sqrt{\frac\pi8}
=\sqrt{\frac{\pi m}{4q\beta}},\qquad
y(\infty)=-\sqrt{\frac{\pi m}{4q\beta}}.
$$
Hence the limiting position is
$$
\boxed{\mathbf r(\infty)=\sqrt{\frac{\pi m}{4q\beta}}\,(1,-1,0).}
$$
The negative $y$ coordinate follows from clockwise rotation of planar velocity for positive charge and positive $B_z$. The position converges through cancellation of increasingly rapid oscillations; its convergence does not mean the constant planar speed tends to zero.
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