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We present a systematic analytic and numerical study of the detection limit of a refractive index sensor employing a directional coupler architecture within a photonic crystal fiber (PCF). The device is based on the coupling between the core mode and a copropagating mode of a satellite waveguide formed by a single hole of the PCF infiltrated by a high-index analyte. Using coupled mode theory as well as full simulations, we investigate the influence of changes in the geometrical parameters of the PCF and the analyte's refractive index on sensor performance, including sensitivity, resonance width, and detection limit. We show that regardless of the details of the sensor's implementation, the smallest detectable refractive index change is inversely proportional to the coupling length and the overlap integral of the satellite mode with the analyte, so that best performance comes at the cost of long analyte infiltration lengths. This is experimentally confirmed in our dip sensor configuration, where the lowest detection limit achievable for realistic implementation is estimated to 7 × 10-8 refractive index units (RIU) based on realistic signal to noise ratios in a commercially available PCF.