The limit of (cos(Pi/2x))^2x as x approaches infinity is 1. This can be proven using the squeeze theorem or by taking the natural logarithm of both sides and using L'Hopital's rule.
To find the limit of (cos(Pi/2x))^2x when x is infinity, you can use the properties of limits such as the squeeze theorem, L'Hopital's rule, or the limit laws. It is important to rewrite the function in a form that makes it easier to evaluate as x approaches infinity.
No, the limit of (cos(Pi/2x))^2x at infinity is not equal to zero. As x approaches infinity, the function approaches 1, not 0. This can be seen by graphing the function or by using the limit definition.
Yes, you can use the substitution method to find the limit of (cos(Pi/2x))^2x at infinity. By substituting a new variable, u, equal to 1/x, the limit can be rewritten as the limit of (cos(Pi/2u))^2u as u approaches 0. From there, you can use techniques such as the squeeze theorem or L'Hopital's rule to evaluate the limit.
The difference between finding the limit of (cos(Pi/2x))^2x as x approaches infinity and finding the limit of (cos(Pi/2x))^2x at infinity is that the former refers to finding the limit as x gets closer and closer to infinity but never actually reaches it, while the latter specifically looks at the limit as x reaches infinity. In other words, the limit at infinity refers to the behavior of the function as x gets larger and larger, while the limit as x approaches infinity refers to the behavior of the function as x gets arbitrarily close to infinity.