Plasma edges in metals typically occur in the visible range, producing characteristic colors of metals. In a lightly doped semiconductor, the plasma edge can occur in the terahertz (THz) frequency range. Due to low scattering rates and variable electron densities in semiconductors, such THz plasma edges can be extremely sharp and greatly tunable. Here, we show that an ultrasharp THz plasma edge exists in a lightly n-doped InSb crystal with a record-high transmittance slope of 80 dB/THz. The frequency at which this sharp edge happens can be readily tuned by changing the temperature, electron density, scattering rate, and sample thickness. The edge frequency exhibited a surprising increase with decreasing temperature below 15 K, which we explain as a result of a weak-to-strong transition in the scattering rate, going from ωτ  ≫ 1 to ωτ ∼ 1. These results indicate that doped narrow-gap semiconductors provide a versatile platform for manipulating THz waves in a controllable manner, especially as a high-pass filter with an unprecedented on/off ratio.