Which pair of functions satisfy f (N) ~g(N)?

Question

I\'ve just started working with algorithms and I am doing some tasks like this question:

I think, the right answer is A. As the functions are the same, or do I miss some

Answer 1

The answer is A. Notice that in that case,

f(N) = N + 2N + 3N = 6N = g(N)

so f(N) ~ g(N).

For the functions given in B, notice that

f(N) = (N + 1) + (N + 2) + (N + 3) = 3N + 6

so the limit of f(N) / g(N) as N tends toward infinity is 3, and therefore f(N) is not tilde of g(N).

For the functions in C, notice that

f(N) / g(N) = 1 / N⁵ + 1 / N⁴ + 1 / N³

and, in the limit, this does not tend to one.

For the functions in D, notice that

f(N) = log N + log 2N + log 3N = log 6N³ = 3 log 6N

so the limit of f(N) over g(N) as N tends toward infinity is 3, so the functions aren't tilde of one another.

Answer 2

Yes, A is the correct answer.

Tilde here means an (approximate, not precise) equivalence relation. Or in other words, the limit of f(N) / g(N) as N approaches infinity = 1.

In answer A, f and g are obviously equivalent, so the limit is 1. This doesn't hold for the other three answers, which have limits 3, 0, and 3 respectively.