112. True, or false? Give reasons for your answers.
a. 1/x⁴ = O(1/x² + 1/x⁴)

Verified step by step guidance

1. Recall the definition of Big-O notation: A function \(f(x)\) is \(O(g(x))\) as \(x \to \infty\) if there exist positive constants \(C\) and \(M\) such that for all \(x > M\), \(|f(x)| \leq C |g(x)|\).
2. Identify the functions involved: \(f(x) = \frac{1}{x^4}\) and \(g(x) = \frac{1}{x^2} + \frac{1}{x^4}\).
3. Analyze the behavior of \(g(x)\) as \(x \to \infty\): Since \(\frac{1}{x^2}\) dominates \(\frac{1}{x^4}\) for large \(x\), \(g(x)\) behaves roughly like \(\frac{1}{x^2}\).
4. Compare \(f(x)\) and \(g(x)\): For large \(x\), \(\frac{1}{x^4}\) is smaller than or equal to a constant multiple of \(\frac{1}{x^2}\), so \(f(x)\) is bounded above by \(g(x)\) times some constant.
5. Conclude whether \(f(x) = O(g(x))\) holds by verifying if the inequality \(|f(x)| \leq C |g(x)|\) can be satisfied for some constants \(C\) and \(M\).

Key Concepts

Big O Notation

Big O notation describes the upper bound of a function's growth rate, showing how it behaves as the input approaches a limit, often infinity. It helps compare functions by their dominant terms, ignoring lower-order terms and constants.

Asymptotic Comparison of Functions

Asymptotic comparison involves analyzing how two functions behave relative to each other as the variable grows large or approaches zero. It determines if one function grows faster, slower, or at the same rate as another.

Properties of Rational Functions

Rational functions are ratios of polynomials. Understanding their behavior, especially for terms like 1/x² and 1/x⁴, is crucial to analyze limits and growth rates, as higher powers in the denominator lead to faster decay as x increases.