Final answer:
To show the limits of T(z) when c = 0 and when c ≠ 0, as well as finding the conditions for T(z) = z, you need to consider the behavior of the numerator and denominator term as z approaches infinity and solve for the variables in the equation az+b/cz+d = z.
Step-by-step explanation:
To show that lim→[infinity] T(z) = [infinity] when c = 0, we need to take the limit as z approaches infinity of the function T(z) = az+b/cz+d. Since c = 0, the denominator becomes 0z+d = d, which approaches d as z approaches infinity. The numerator remains az+b. As z becomes larger, the numerator also becomes larger, leading to a result of [infinity].
To show that lim→[infinity] T(z) = a/c and lim→₋ T(z) = [infinity] when c ≠ 0, we can rewrite the function T(z) as T(z) = (az+b/cz+d) = (a/c + b/cz+d/c), which can be simplified further as T(z) = a/c + b/cz/(1+d/cz). As z approaches infinity, the second term in the expression becomes negligible, resulting in T(z) ≈ a/c. As z approaches negative infinity, the first term dominates and the second term approaches 0, resulting in T(z) ≈ [infinity].
To find the conditions on a, b, c, and d for T(z) = z, we can equate the function T(z) to z and solve for the variables. The equation becomes az+b/cz+d = z. Simplifying further, we get az+b = cz²+dz. Comparing the coefficients of the quadratic equation, we have a = c, b = d, and 0 = 0. Therefore, the conditions for T(z) = z are a = c, b = d, and any value for a and b.