Determine the following limits.
lim x→∞ (x⁴ − 1) / (x⁵ + 2)

Verified step by step guidance

1. Identify the highest power of x in both the numerator and the denominator. Here, the highest power in the numerator is x^4 and in the denominator is x^5.
2. Divide every term in the numerator and the denominator by x^5, the highest power of x in the denominator.
3. Rewrite the expression: (x^4/x^5 - 1/x^5) / (x^5/x^5 + 2/x^5).
4. Simplify the expression: (1/x - 1/x^5) / (1 + 2/x^5).
5. Evaluate the limit as x approaches infinity. As x approaches infinity, terms with x in the denominator approach zero, simplifying the expression to 0/1.

Key Concepts

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches infinity. This concept is crucial for understanding how functions behave for very large values of x, which can help determine horizontal asymptotes and the end behavior of polynomial functions.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. In the given limit, both the numerator and denominator are polynomials, and their degrees play a significant role in determining the limit's value as x approaches infinity.

Dominant Term

The dominant term in a polynomial is the term with the highest degree, which significantly influences the function's behavior as x approaches infinity. In the limit provided, identifying the dominant terms in both the numerator and denominator allows for simplification and accurate evaluation of the limit.