Final answer:
The limit of the given function as x approaches infinity is 0.
Step-by-step explanation:
To find the limit as x approaches infinity of the given function, we need to divide the highest power of x in the numerator by the highest power of x in the denominator. In this case, the highest power of x is x³ in both the numerator and denominator. Dividing the coefficients of x³, we get the limit as x approaches infinity of (4 + 3x - 8/x²) / (5x³ + x²). As x approaches infinity, both the numerator and denominator grow towards infinity, so the limit is
L = (lim as x approaches infinity of 4 + 3x - 8/x²) / (lim as x approaches infinity of 5x³ + x²).
Since the exponent of x² in the denominator is greater than the exponent of x in the numerator, as x approaches infinity, the term 8/x² goes to zero. Therefore, the numerator simplifies to 4 + 3x. Similarly, the term x² in the denominator can be neglected relative to the term x³ as x approaches infinity. Therefore, the denominator simplifies to 5x³. The limit now becomes
L = (lim as x approaches infinity of 4 + 3x) / (lim as x approaches infinity of 5x³).
Taking the limits separately, we get L = (4 + 3 * infinity) / (5 * infinity³). Since infinity is not a real number, we can write this as L = (4 + infinity) / infinity³. It can be seen that infinity³ is much larger than infinity, so we can neglect the term 4 in the numerator relative to infinity³. Therefore, the limit simplifies to L = infinity / infinity³ = 1 / infinity² = 0.