A paradox in Taylor Polynomials?

In Taylor's formula with remainder, I know that f(x) = P_n (x) + R_n (x), where P_n (x) is the nth-degree Taylor polynomial of f at c and R_n (x) is the Taylor remainder. If lim n->oo R_n (x) = 0, then as n increases, P_n (x) -> f(x); hence the approximation formula f(x) ~ P_n (x) improves as n gets larger. So, let's say we use the first two nonzero terms of the Maclaurin series for sin x to obtain the approximation formula sin x ~ x - x^3/3!. We could write the expression on the right as P_3 (x). As n increases, R_n (x) gets more closer to 0 and the maximum approximation error will also gets closer to 0.