Prove that lim (x->0) x^4 cos 2/x = 0.

To prove that the limit of the function as x approaches 0 is 0, we can use the squeeze theorem. The squeeze theorem states that if we can find two functions that "squeeze" our given function, and if both of those functions have a limit of 0 as x approaches the same point, then our given function must also have a limit of 0 at that point.

In this case, our given function is f(x) = x^4 * cos(2/x). First, let's find two functions that squeeze f(x) as x approaches 0.

Consider the function g(x) = x^4. As x approaches 0, the function g(x) also approaches 0. This is because any number raised to the power of 4 will become infinitesimally small as x approaches 0.

Now consider the function h(x) = cos(2/x). Here, we need to show that h(x) is bounded between two functions that approach 0. Since the cosine function is bounded between -1 and 1 for all real values, we have -1 ≤ cos(2/x) ≤ 1 for all x. Now, let's multiply this inequality by x^4:

-1 * x^4 ≤ x^4 * cos(2/x) ≤ 1 * x^4

This becomes:

-x^4 ≤ f(x) ≤ x^4

As x approaches 0, both -x^4 and x^4 approach 0. Therefore, f(x) is squeezed between two functions that approach 0. By the squeeze theorem, the limit of f(x) as x approaches 0 must also be 0.

Hence, we have proved that lim(x->0) x^4 * cos(2/x) = 0.

lim (x->0) x^4 cos 2/x=

=lim (x->0) x^4 * lim (x>0) cos 2/x

= 0 * lim (x>0) cos 2/x

cos (a) will always be between -1 and 1. but this is irrelevant since x^4 is 0 at x=0, and anything multiplied by 0 is 0

:P