Evaluate the limit as x approaches infinity of (2x^3 + x^2)/(5x^3 - 2).

Evaluate the limit as x approaches infinity of (2x^3 + x^2)/(5x^3 - 2).

• A. 0
• B. 1
• C. 2/5
• D. -2/5

Answer: (C)

When x grows without bound, the highest power terms dominate a rational function. Here both numerator and denominator have x^3 as their leading term, so the limit is guided by the leading coefficients: 2x^3 in the numerator and 5x^3 in the denominator, giving 2/5. A precise way to see this is to divide top and bottom by x^3: (2 + 1/x) / (5 - 2/x^3). As x → ∞, 1/x → 0 and 2/x^3 → 0, leaving 2/5. So the limit equals 2/5.

When x grows without bound, the highest power terms dominate a rational function. Here both numerator and denominator have x^3 as their leading term, so the limit is guided by the leading coefficients: 2x^3 in the numerator and 5x^3 in the denominator, giving 2/5.

A precise way to see this is to divide top and bottom by x^3: (2 + 1/x) / (5 - 2/x^3). As x → ∞, 1/x → 0 and 2/x^3 → 0, leaving 2/5.

So the limit equals 2/5.