Diverse cell types employ mechanisms to maintain size homeostasis and minimize aberrant fluctuations in cell size. It is well known that exponential cellular growth can drive unbounded intercellular variations in cell size, if the timing of cell division is size independent. Hence coupling of division timing to size is an essential feature of size control. We formulate a stochastic model, where exponential cellular growth is coupled with random cell division events, and the rate at which division events occur increases as a power function of cell size. Interestingly, in spite of nonlinearities in the stochastic dynamical model, statistical moments of the newborn cell size can be determined in closed form, providing fundamental limits to suppression of size fluctuations. In particular, formulas reveal that the magnitude of fluctuations in the newborn size is determined by the inverse of the size exponent in the division rate, and this relationship is independent of other model parameters, such as the growth rate. We further expand these results to consider randomness in the partitioning of mother cell size among daughters at the time of division. The sensitivity of newborn size fluctuations to partitioning noise is found to monotonically decrease, and approach a non-zero value, with increasing size exponent in the division rate. Finally, we discuss how our analytical results provide limits on noise control in commonly used models for cell size regulation.