Final answer:
The limit of tan(1/n)/n1/2 as n approaches infinity converges to 0. L'Hôpital's Rule can be applied since the initial form is indeterminate, and after taking derivatives and simplification, it is found that the limit converges.
Step-by-step explanation:
The student is asking whether the limit of the function tan(1/n)/n1/2 converges as n approaches infinity. To determine this, we can use L'Hôpital's Rule which states that if the limit of f(n)/g(n) results in an indeterminate form of 0/0 or ∞/∞, then the limit of f(n)/g(n) as n approaches a point can be found by taking the limit of their derivatives instead.
L'Hôpital's rule would apply if we reexpress the function as tan(x)/x where x = 1/n. As n approaches infinity, x approaches 0, rendering the limit to be in the form of 0/0. Differentiating the numerator and denominator, we get the limit of sec2(x) as x approaches 0, and that equals 1 since sec(0) = 1. This differentiation, in turn, simplifies the original limit to 1/n1/2 which converges to 0 as n approaches infinity. Therefore, the limit does converge, and the answer is (a) Yes.