Question

Find the least integer n such that f(x) is o(xn) for each of these functions. a) f(x) = 2x² x³ log x b) f(x) = 3x⁵ (log x)⁴ c) f(x) = (x⁴ x² 1)/(x⁴ 1) d) f(x) = (x³ 5 log x)/(x⁴ 1)

Answer 1

To find the least integer n such that f(x) is o(xn) for each function, we need to determine the highest power of x in each function and add 1 to it.

Step-by-step explanation:

To find the least integer n such that f(x) is o(xn) for each function, we need to determine the highest power of x in each function and add 1 to it. Let's go through each function:

a) f(x) = 2x² x³ log x

The highest power of x in this function is x³. Therefore, n = 3 + 1 = 4.

b) f(x) = 3x⁵ (log x)⁴

The highest power of x in this function is x⁵. Therefore, n = 5 + 1 = 6.

c) f(x) = (x⁴ x² 1)/(x⁴ 1)

The highest power of x in this function is x⁴. Therefore, n = 4 + 1 = 5.

d) f(x) = (x³ 5 log x)/(x⁴ 1)

The highest power of x in this function is x⁴. Therefore, n = 4 + 1 = 5.

So the least integer n such that f(x) is o(xn) for each function is 4.