To determine the stability limit of the given servo-mechanism, we need to analyze the open-loop transfer function:

G(s)H(s) = \frac{K_a(2+sT_1)}{s^2(1+sT_2)}

Stability of a servo-mechanism can be analyzed by the Routh-Hurwitz criterion. This criterion involves ensuring that all coefficients of the characteristic equation derived from the denominator have positive values.

Step-by-Step Analysis:

First, we need to find the characteristic equation from the given transfer function. The characteristic equation is obtained by setting the denominator of the closed-loop transfer function to zero.

The closed-loop transfer function has the form:

1 + G(s)H(s) = 0

This results in:

1 + \frac{K_a(2+sT_1)}{s^2(1+sT_2)} = 0

This implies:

s^2(1+sT_2) + K_a(2+sT_1) = 0

Expanding this, we get:

s^2 + s^3T_2 + 2K_a + K_asT_1 = 0

The characteristic equation is:

s^3T_2 + s^2 + K_asT_1 + 2K_a = 0

From the Routh-Hurwitz criterion for a third-order system, for stability, certain conditions must be met.

Condition Analysis:

By examining the coefficients of s terms, we observe the following stability condition from the characteristic equation's coefficients:

• The coefficient of each power of s must be positive.
• Moreover, after applying the Routh table, the stability condition for this specific setup often boils down to considering the inequality involving T_1 and T_2.

In this specific case, it has been derived that:

T_1 > 2T_2

is the criterion to ensure that all necessary conditions for stability in the Routh Array are satisfied.

Conclusion:

Therefore, among the given options, the correct stability condition is:

T_1 > 2T_2