Final answer:
To find the limit of (2 + sin(n))^(1/n) using the sandwich theorem, we need to find two other functions whose limits we can easily determine and that sandwich the given function. The sandwich theorem states that if a function f(x) is always between two other functions g(x) and h(x) for values of x close to a particular point, and if the limits of g(x) and h(x) as x approaches that point are both equal to L, then the limit of f(x) as x approaches the point is also L.
Step-by-step explanation:
To find the limit of (2 + sin(n))^(1/n) using the sandwich theorem, we need to find two other functions whose limits we can easily determine and that sandwich the given function.
The sandwich theorem states that if a function f(x) is always between two other functions g(x) and h(x) for values of x close to a particular point, and if the limits of g(x) and h(x) as x approaches that point are both equal to L, then the limit of f(x) as x approaches the point is also L.
In this case, we can see that -1 ≤ sin(n) ≤ 1 for all n. Therefore, -1 ≤ 2 + sin(n) ≤ 3. Taking the nth root of these inequalities, we have -1 ≤ (2 + sin(n))^(1/n) ≤ 3^(1/n).
As n approaches infinity, both -1 and 3^(1/n) approach 1. Therefore, by the sandwich theorem, the limit of (2 + sin(n))^(1/n) as n approaches infinity is also 1.