Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ ∞ (4x³ - 2x² + 6) / (πx³ + 4)

Verified step by step guidance

1. First, identify the form of the limit as x approaches infinity. The given expression is (4x³ - 2x² + 6) / (πx³ + 4). As x approaches infinity, both the numerator and the denominator approach infinity, which is an indeterminate form ∞/∞.
2. Since the limit is in the indeterminate form ∞/∞, we can apply l'Hôpital's Rule. This rule states that if the limit of f(x)/g(x) as x approaches a value is in the form 0/0 or ∞/∞, then it can be evaluated as the limit of f'(x)/g'(x), provided this new limit exists.
3. Differentiate the numerator and the denominator separately. The derivative of the numerator 4x³ - 2x² + 6 is 12x² - 4x. The derivative of the denominator πx³ + 4 is 3πx².
4. Now, apply l'Hôpital's Rule by taking the limit of the new expression: lim_x→∞ (12x² - 4x) / (3πx²).
5. Simplify the expression by dividing each term by x², the highest power of x in the denominator. This gives lim_x→∞ (12 - 4/x) / (3π). As x approaches infinity, the term 4/x approaches 0, simplifying the limit to 12 / 3π.

Key Concepts

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior at points where it may not be explicitly defined, such as at infinity or at points of discontinuity. Evaluating limits is crucial for determining the continuity and differentiability of functions.

l'Hôpital's Rule

l'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. The rule states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This process can be repeated if the result remains indeterminate.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to whole number powers and their coefficients. In the context of limits, the degree of the polynomial in the numerator and denominator plays a crucial role in determining the limit as x approaches infinity. The leading terms of these polynomials dominate the behavior of the function at extreme values, simplifying the limit evaluation.