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

Guida verificata passo dopo passo

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 \(x > M\), \(|f(x)| \leq C |g(x)|\).

Identify the functions involved: \(f(x) = \frac{1}{x^4}\) and \(g(x) = \frac{1}{x^2} + \frac{1}{x^4}\).

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 \(\frac{1}{x^2}\).

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.

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 \(M\).

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