How to find the bandwidth of a function having infinite DC gain?
How do you define the term &quot;bandwidth&quot; in this case?
There are some transfer functions, which cannot be described sufficiently using a &quot;bandwidth&quot;.
Example: What is the bandwidth for an integrating device? We only can specify the region where the integration function fulfills our accuracy reqirements (because, in reality, the phase shift of 90 deg does exist at one single frequency only).
I rather think that the term &quot;bandwidth&quot; is used as a parameter that best describes what the circuit &quot;should do&quot; within a certain frequency region (to pass or to stop/attenuate some frequencies). Therefore, your question can be answered only when we know something about the purpose and the tasks to be fulfilled by the given transfer function and the corresponding circuit, respectively.
EDIT (added): As another example, consider a BESSEL/THOMSON lowpass. This a lowpass filter with comparable bad attenuation properties, but with a nearly constant group delay within a certain frequency range. Therefore, such a filter will be used primarily because of its behaviour in the time domain (undistorted pulse transmission) - and the &quot;bandwidth&quot; (passband) is specified very often with respect to the allowed (acceptable) group delay variations.