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Signal Cutoff Gain (Electrical & Computer Engineering)

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A digital signal processing (DSP) engineer is designing an ideal brick-wall low-pass filter. The normalized voltage gain G(f)G(f) of the filter as a function of the signal frequency ff (in kHz) is modeled by:

G(f)=f2−16f−4G(f) = \frac{f^2 - 16}{f - 4}

At the cutoff frequency of f=4 kHzf = 4\text{ kHz}, the filter's electronic circuitry experiences a singularity (direct evaluation yields 0/00/0, making the exact gain value G(4)G(4) undefined).

  1. Explain why direct evaluation at f=4 kHzf = 4\text{ kHz} fails.
  2. Use a numerical approach to show that as the input frequency approaches the cutoff f=4 kHzf = 4\text{ kHz} from the left (f→4−f \to 4^-) and right (f→4+f \to 4^+), the gain approaches a stable value. What is this limit?