What is the infinitesimal order of x ^ 1 / 3 + x ^ 4 / 3, x ^ 1 / 2 and 1-cos x ^ 2 when the ball x approaches 0?

What is the infinitesimal order of x ^ 1 / 3 + x ^ 4 / 3, x ^ 1 / 2 and 1-cos x ^ 2 when the ball x approaches 0?

When x ^ 3 is infinitesimal (x 1 / 3 + x 1 / 3), X is infinitesimal
X ^ 1 / 2 is obviously infinitesimal of order 1 / 2 of X
1-cosx ^ 2 = 1 - [1 - (x ^ 2) ^ 2 / 2 + O (x ^ 4)] = x ^ 4 / 2 + O (x ^ 4), so it is infinitesimal of order 4 of X

Using the property of infinitesimal, the following limit Lim SiNx Sina / x-a (x →∞) is calculated
There is another way:
lim(1-x^2)/sinπx x→1

Because | SiNx|

Why is 1-cosx and (x ^ 2) / 2 equivalent infinitesimals when x tends to zero?

Because these two numbers also tend to zero, we can get the quadratic term by Fourier transform of 1-cosx, which is equivalent to (x ^ 2) / 2