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In this paper, we consider the problem of the accuracy of estimating the location and other attributes of a moving single molecule whose trajectory is imaged by fluorescence microscopy. As accuracy in parameter estimation is closely related to the Fisher information matrix, we first give a general expression of the Fisher information matrix for the estimated parameters for a single object moving in three-dimensional (3D) space. Explicit Cramér-Rao lower bound (CRLB) expressions are then obtained from the Fisher information matrix for a single object moving in the two-dimensional (2D) focus plane with the object trajectory being either linear or circular. We also investigate how extraneous noise sources, pixelation, parameters of the detection system and parameters of the trajectory affect the limit of the accuracy. The results obtained in this paper provide insights that enable the experimentalists to optimize their experimental setups for tracking single molecules in order to achieve the best possible accuracy. They are also applicable to the general problem of tracking an object using quantum limited detectors.