A theorem states that "If lim(x->infinity) f(x)=L, then the sequence a_n=f(n) converges and lim(n->infinity) a_n=L." Show that the converse is false. In other words, find a function f(x) such that a_n=f(n) converges but lim(x->infinity) f(x) does not exist. Thank you!
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f(x) = sin(πx)
f(1) = f(2) = f(3) = ... = f(n) = 0, so lim n->∞ sin(nπ) = 0, which converges
On the other hand,
lim x->∞ sin(πx) oscillates between -1 and +1, hence is divergent
f(1) = f(2) = f(3) = ... = f(n) = 0, so lim n->∞ sin(nπ) = 0, which converges
On the other hand,
lim x->∞ sin(πx) oscillates between -1 and +1, hence is divergent