If the range of t is bounded above and below then there is such a polynomial by the Weierstrass Approximation Theorem. If t is allowed to be indefinitely large, positive or negative, then there is not for the reason you note.
> True or False? : > > There exists a polynomial P such that: > > | P(t) - cos(t) | <= 10^-6. > > |t|, meaning the absolute value of t. > > I said false. Because P(t) could be very large. Then cos(t) is > comparatively small. Then the answer is a large positive number, > greater than 10^-6. > > Although then I started to think along the lines of the Taylor series, > because > > cos(t) = 1 - x^2/2! + x^4/4! - x^6/6! + ... > > and this is valid for all values of t, so perhaps there could be a > polynomial (I'm not sure what that would be). > > So on the other hand it could also be true. Which is the correct > answer? > > Neenag@cableol.co.uk