Final answer:
Increasing the gain, K, in an open loop transfer function makes the poles of the closed-loop system move towards the zeros of the open-loop transfer function. This is a result of the feedback loop influencing the system's response as K changes. As K goes to infinity, the effect of the denominator reduces, and poles converge on the zeros.
Step-by-step explanation:
When discussing the open loop transfer function G(s) = (s+4)(s+5)/(s+2)(s+6) and the effects of increasing the gain, K, towards infinity, the roots of the characteristic equation will shift in the complex plane. As K increases, the poles (roots of the denominator of the transfer function when the feedback loop is closed with gain K) of the closed-loop system will move towards the zeros of the open-loop transfer function (roots of the numerator), and since the system does not have any zeros on the right half of the complex plane, the poles will either remain in the left half or move onto the imaginary axis if the system becomes marginally stable.
Why does this happen? In control theory, a system's stability is determined by the location of its poles in the complex plane. An increasing gain can cause the poles to move, potentially leading towards instability. The reason the poles move towards the zeros is due to the effect of the feedback loop, which modifies the system's response as the gain changes. As K goes to infinity, if the open-loop system is stable, the closed-loop poles converge on the open-loop zeros because the effect of the denominator (which defines the poles when K is finite) diminishes relative to the numerator as K increases.