Autonomous Vehicle Radar Distance Margin (Computer Engineering)

In an autonomous vehicle driver-assistance system, a computer engineering team develops a dual-frequency frequency-modulated continuous-wave (FMCW) radar algorithm to estimate target tracking separation distance. The spatial calibration offset $\Delta d(x)$ (in meters) between direct reflection and synthetic aperture reconstruction as longitudinal range $x$ (in meters) expands is modeled by:

As the vehicle tracks obstacles at high speeds over long highway distances ($x \to \infty$), the team needs to know if the spatial offset diverges, vanishes, or settles to a constant calibration bias.

1. Explain why direct substitution of $x \to \infty$ results in an indeterminate form.
2. Evaluate the limit of the distance offset analytically using radical conjugate rationalization:
   $$\lim_{x \to \infty} \left( \sqrt{x^2 + 8x} - x \right)$$
3. State the horizontal asymptote and explain how this constant offset is programmed into the vehicle's collision-avoidance radar firmware.