Evaluate the following limits in two different ways: with and without l’Hôpital’s Rule.
lim_x→∞ (2x⁵ - x + 1) / (5x⁶ + x)

Verified step by step guidance

1. First, let's evaluate the limit without using l'Hôpital's Rule. Identify the highest power of x in both the numerator and the denominator. In the numerator, the highest power is x^5, and in the denominator, it is x^6.
2. Divide every term in the numerator and the denominator by x^6, the highest power of x in the denominator. This gives us: (2x^5/x^6 - x/x^6 + 1/x^6) / (5x^6/x^6 + x/x^6).
3. Simplify the expression: (2/x - 1/x^5 + 1/x^6) / (5 + 1/x^5). As x approaches infinity, terms with x in the denominator approach zero.
4. Now, evaluate the limit: lim_{x→∞} (2/x - 1/x^5 + 1/x^6) / (5 + 1/x^5) = 0/5 = 0.
5. Next, let's use l'Hôpital's Rule. Since the limit is in the indeterminate form ∞/∞, differentiate the numerator and the denominator: d/dx(2x^5 - x + 1) = 10x^4 - 1, and d/dx(5x^6 + x) = 30x^5 + 1. Evaluate the new limit: lim_{x→∞} (10x^4 - 1) / (30x^5 + 1). Repeat l'Hôpital's Rule if necessary until the limit is no longer indeterminate.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, we are interested in the limit as x approaches infinity, which helps us understand the behavior of the function at extreme values. Evaluating limits is crucial for determining continuity, derivatives, and integrals.

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 separately. This rule simplifies the process of finding limits, especially for rational functions.

Polynomial Growth Rates

Understanding polynomial growth rates is essential for evaluating limits involving polynomials. In the given limit, the degrees of the polynomials in the numerator and denominator determine the limit's value as x approaches infinity. Generally, the term with the highest degree dominates the behavior of the polynomial, allowing for simplification when calculating limits.