Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (3x⁴ - x²) / (6x⁴ + 12) 

Verified step by step guidance

1. First, identify the form of the limit as x approaches infinity. The given expression is (3x⁴ - x²) / (6x⁴ + 12). As x approaches infinity, both the numerator and the denominator approach infinity, indicating an indeterminate form ∞/∞.
2. Since the limit is in the indeterminate form ∞/∞, l'Hôpital's Rule can be applied. According to l'Hôpital's Rule, if the limit of f(x)/g(x) as x approaches a value is in the form 0/0 or ∞/∞, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator.
3. Differentiate the numerator: The derivative of 3x⁴ is 12x³, and the derivative of -x² is -2x. Therefore, the derivative of the numerator is 12x³ - 2x.
4. Differentiate the denominator: The derivative of 6x⁴ is 24x³, and the derivative of the constant 12 is 0. Therefore, the derivative of the denominator is 24x³.
5. Apply l'Hôpital's Rule: Evaluate the limit of the new expression (12x³ - 2x) / 24x³ as x approaches infinity. Simplify the expression and determine the limit by considering the highest degree terms in the numerator and denominator.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches infinity helps determine the behavior of the function at extreme values.

l'Hôpital's Rule

l'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if these forms occur, the limit of the ratio of two functions can be found by taking the derivative of the numerator and the derivative of the denominator. This rule simplifies the process of finding limits in complex expressions.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number powers, combined using addition, subtraction, and multiplication. In the given limit, the degrees of the polynomials in the numerator and denominator determine the limit's behavior as x approaches infinity. Understanding the leading terms of these polynomials is crucial for evaluating the limit.