Determine the following limits. 
lim x→−∞ (2x^-8 + 4x³)

Verified step by step guidance

1. Identify the dominant term in the expression as \(x\) approaches \(-\infty\). The expression is \(2x^{-8} + 4x^3\).
2. Since \(x^3\) grows faster than \(x^{-8}\) as \(x\) approaches \(-\infty\), the term \(4x^3\) is dominant.
3. Rewrite the expression focusing on the dominant term: \(4x^3\).
4. Consider the behavior of \(4x^3\) as \(x\) approaches \(-\infty\).
5. Conclude that the limit of the expression is determined by the behavior of the dominant term \(4x^3\) as \(x\) approaches \(-\infty\).

Key Concepts

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches positive or negative infinity. This concept is crucial for understanding how functions behave in extreme cases, allowing us to determine whether they approach a specific value, diverge, or oscillate.

Dominant Terms

In polynomial expressions, the dominant term is the term with the highest degree, which significantly influences the function's behavior as x approaches infinity or negative infinity. Identifying the dominant term helps simplify the limit calculation by focusing on the most impactful part of the expression.

Polynomial Growth Rates

Polynomial growth rates refer to how different polynomial terms grow relative to each other as x approaches infinity or negative infinity. Understanding that higher-degree terms grow faster than lower-degree ones is essential for evaluating limits, especially when combining terms of varying degrees.