Evaluate each limit and justify your answer. 
lim x→0 (x^8−3x^6−1)^40

Verified step by step guidance

1. Identify the limit expression: \(\lim_{x \to 0} (x^8 - 3x^6 - 1)^{40}\).
2. Evaluate the expression inside the limit as \(x\) approaches 0: \(x^8 - 3x^6 - 1\). Substitute \(x = 0\) to get \$0^8 - 3(0)^6 - 1 = -1$.
3. Since the expression inside the limit approaches a constant value, \(-1\), as \(x\) approaches 0, the limit simplifies to \((-1)^{40}\).
4. Recognize that \((-1)^{40}\) is a power of \(-1\). Since 40 is an even number, \((-1)^{40}\) equals 1.
5. Conclude that the limit is 1, as the expression simplifies to a constant value.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of the function as x approaches 0. Understanding limits is crucial for evaluating functions that may not be directly computable at specific points.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The expression in the limit, (x^8−3x^6−1), is a polynomial function. Analyzing polynomial functions helps in determining their behavior at specific points, such as identifying leading terms and their contributions to the limit.

Continuous Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. In evaluating the limit of the given polynomial raised to a power, we can apply the property of continuity, which allows us to substitute the limit value directly into the function, simplifying the evaluation process.