Function Growth Explanation

To rank the given functions according to their growth rates as n→∞, we need to consider their asymptotic behavior. Here’s how they compare, from slowest to fastest growth:

1. log⁡²⁰²¹(n): This function grows very slowly compared to others. It’s a polynomial in log⁡(n)\log(n)log(n), and the base of the logarithm is usually not specified in asymptotic analysis, but it doesn’t affect the growth rate significantly in this context.
2. sqrt(n​): This function grows faster than any logarithmic function but slower than linear or polynomial functions.
3. nlog⁡²(n) : This grows faster than sqrt(n) because it combines linear growth with a logarithmic factor. However, it grows slower than any polynomial function of nnn.
4. n²: This is a polynomial function of nnn and grows faster than nlog⁡²(n).
5. sqrt(2ⁿ)​: This can be simplified to 2^n/2, which grows exponentially but slower than n² and nⁿ
6. n!: This factorial function grows faster than exponential functions like 2ⁿ. Factorials grow faster than any polynomial or exponential function for sufficiently large n.
7. nⁿ: This function grows the fastest among the given functions. It’s an exponential function with the base and exponent being n, which grows extraordinarily fast.