Final answer:
To find the least integer n such that f(x) is O(x^n) for each function, we need to determine how the function grows or decays as x increases. The least integer n for f(x) = √x is 0, for f(x) = e^x is any positive integer, and for f(x) = ln(x) is any positive real number.
Step-by-step explanation:
To find the least integer n such that f(x) is O(x^n) for each function, we need to determine how the function grows or decays as x increases. Let's analyze each function separately:
a) f(x) = √x
The function √x represents the square root of x. As x increases, the value of √x increases as well, but at a decreasing rate. This suggests that the function √x can be approximated by O(x^0) or O(1). Therefore, the least integer n is 0.
b) f(x) = e^x
The function e^x represents exponential growth. As x increases, the value of e^x increases rapidly. This suggests that the function e^x can be approximated by O(x^n) for any positive integer n. Therefore, the least integer n is any positive integer.
c) f(x) = ln(x)
The function ln(x) represents logarithmic growth. As x increases, the value of ln(x) increases slowly. This suggests that the function ln(x) can be approximated by O(x^n) for any positive real number n. Therefore, the least integer n is any positive real number.