<p>If \(g > p > 0\) then \(\lim_{x \to \infty} \frac{dx^p + ex^{p-2} + c}{dx^q + ex^{q-2} + b}\) is</p>
Step-by-Step Solution
Key Concept: When taking limits of rational functions at infinity, the behavior is completely determined by comparing the degrees of the numerator and denominator polynomials. The highest degree terms dominate, and lower degree terms become negligible.
<p><strong>Step 1:</strong> Identify the degrees of numerator and denominator polynomials.</p><p>Numerator: highest degree term is dx^p (degree = p)</p><p>Denominator: highest degree term is dx^q (degree = q, where q > p given)</p><p><strong>Step 2:</strong> Apply the rule for limits of rational functions as x → ∞.</p><p>Since the degree of denominator (q) > degree of numerator (p), divide both numerator and denominator by x^q:</p><p>$$\lim_{x \to \infty} \frac{dx^p + ex^{p-2} + c}{dx^q + ex^{q-2} + b} = \lim_{x \to \infty} \frac{\frac{d}{x^{q-p}} + \frac{e}{x^{q-p+2}} + \frac{c}{x^q}}{d + \frac{e}{x^2} + \frac{b}{x^q}}$$</p><p><strong>Step 3:</strong> As x → ∞, all terms with x in the denominator approach 0.</p><p>$$= \frac{0}{d} = 0$$</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: D