Limits and Infinity


Find the limits in Exercises 37–46.


x⁴ + x³
lim -----------------
x→∞ 12x³ + 128

Verified step by step guidance

1. First, identify the highest power of x in both the numerator and the denominator. In this case, the highest power in the numerator is x^4 and in the denominator is x^3.
2. To simplify the expression, divide every term in the numerator and the denominator by x^3, the highest power of x in the denominator.
3. The expression becomes: (x^4/x^3 + x^3/x^3) / (12x^3/x^3 + 128/x^3). Simplify each term: x + 1 / (12 + 128/x^3).
4. As x approaches infinity, the term 128/x^3 approaches 0 because any constant divided by an infinitely large number approaches 0.
5. The limit simplifies to: lim (x + 1) / 12 as x approaches infinity. Since x is in the numerator and grows without bound, the limit is infinity.