9. True, or false? As x→∞,
c. x = O(x+5)

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 \(x_0\) such that for all \(x > x_0\), \(|f(x)| \leq C |g(x)|\).
2. Identify the functions in the problem: Here, \(f(x) = x\) and \(g(x) = x + 5\).
3. Analyze the behavior of \(f(x)\) and \(g(x)\) as \(x \to \infty\): Since \(x + 5\) behaves like \(x\) for large \(x\), the two functions grow at the same rate.
4. Set up the inequality to check if \(x = O(x + 5)\): We want to find constants \(C\) and \(x_0\) such that \(x \leq C (x + 5)\) for all \(x > x_0\).
5. Consider dividing both sides by \(x + 5\) (which is positive for large \(x\)) to get \(\frac{x}{x + 5} \leq C\). Since \(\lim_{x \to \infty} \frac{x}{x + 5} = 1\), we can choose \(C\) slightly larger than 1 and find an \(x_0\) to satisfy the inequality.

Key Concepts

Big O Notation

Big O notation describes an upper bound on the growth rate of a function as its input approaches infinity. It is used to compare the asymptotic behavior of functions, indicating that one function grows no faster than another up to a constant multiple.

Asymptotic Behavior of Functions

Asymptotic behavior studies how functions behave as the input becomes very large. Understanding this helps determine if one function can be bounded by another, which is essential for applying Big O notation correctly.

Comparison of Linear Functions

When comparing linear functions like x and x+5, the constant term becomes insignificant as x approaches infinity. This means x and x+5 grow at the same rate asymptotically, which is key to evaluating statements involving Big O notation.