has infinitely many eigenvalues
and indicate the behaviour of lambda as n goes to infinity.
You really ought not to need much of a hint in order to do this. You should recognise as a simple harmonic motion equation (assuming that ), with solution , where . The initial conditions will give you a condition on , of the form . This is not an equation that you can solve explicitly, but by drawing a graph of the tan function, you can see that there is a doubly-infinite family of solutions, which for large values of will be close to odd multiples of .