10. True, or false? As x→∞,
a. 1/(x+3) = O(1/x)

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: \(f(x) = \frac{1}{x+3}\) and \(g(x) = \frac{1}{x}\).
3. Compare the behavior of \(f(x)\) and \(g(x)\) as \(x \to \infty\). Notice that \(\frac{1}{x+3}\) behaves very similarly to \(\frac{1}{x}\) because adding 3 becomes insignificant for very large \(x\).
4. To verify the Big-O relationship, consider the ratio \(\frac{|f(x)|}{|g(x)|} = \frac{\frac{1}{x+3}}{\frac{1}{x}} = \frac{x}{x+3}\). As \(x \to \infty\), this ratio approaches 1, which is a finite constant.
5. Since the ratio approaches a finite constant, there exist constants \(C\) and \(x_0\) such that \(|f(x)| \leq C |g(x)|\) for all \(x > x_0\), confirming that \(\frac{1}{x+3} = O\left(\frac{1}{x}\right)\) is true.

Key Concepts

Big O Notation

Big O notation describes the upper bound of a function's growth rate, focusing on its behavior as the input approaches infinity. It provides a way to classify functions according to their dominant terms, ignoring constant factors and lower-order terms.

Asymptotic Behavior of Functions

Asymptotic behavior studies how functions behave as the input variable grows very large. Understanding limits and dominant terms helps compare functions like 1/(x+3) and 1/x as x approaches infinity.

Equivalence of Functions in Big O Terms

Two functions f(x) and g(x) are related by f(x) = O(g(x)) if there exist constants C and x0 such that |f(x)| ≤ C|g(x)| for all x > x0. This means f(x) grows no faster than a constant multiple of g(x) for large x.