Final answer:
To determine the value of 'a' for the given limit as x approaches negative infinity, the leading term ax² of the square root dominates, which simplifies to -ax when taking the negative square root. Thus, the value of a satisfies a = -d for the limit to equal 2.
Step-by-step explanation:
To find the value of a for which the limit Lim x → ∞ (sqrt(ax² + bx + c))/d - ax = 2 holds true, we can investigate the behavior of the expression as x approaches negative infinity. The term with the highest power of x in the square root will dominate the behavior of the expression, which is ax². Thus, the term sqrt(ax²) will be analogous to |ax|. Since we are considering the limit as x approaches negative infinity, we need to take the negative square root, i.e., -ax.
Therefore, the expression simplifies to: lim x → ∞ (-ax/d - ax) = -2ax/d. Setting this equal to 2 and solving for a, we get a = -d.
Please note that while the quadratic equation is mentioned in the reference, it is not necessarily relevant to solving this limit problem, except to highlight the effect of the leading coefficient on the behavior of the quadratic expression.