Define
T(x)=x2−1x+3−2−(2x+1−3).
The variable x is restricted to values close to 1 but not equal to 1.
a. Show, through a systematic sequence of algebraic transformations, how the numerator of T(x) can be rewritten so that every occurrence of a radical is eliminated from the top-level expression. Clearly indicate any factorable terms that emerge.
b. After completing part (a), evaluate x→1limT(x). Provide a step-by-step justification for your result, including the treatment of all factors that cancel or remain finite at the limit point.