Define T(x)= x+3 -2 ; - ; ( 2x+1 - 3 ) x^ 2 -1

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Define T(x)=x+3−2  −  (2x+1−3)x2−1.T(x)=\frac{\sqrt{x+3}-2\; -\;\bigl(\sqrt{2x+1}-\sqrt{3}\bigr)}{x^{2}-1}. The variable xx is restricted to values close to 11 but not equal to 11. (1) Show, through a systematic sequence of algebraic transformations, how the numerator of T(x)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. (2) After completing part (a), evaluate lim⁡x→1T(x)\displaystyle \lim_{x\to 1}T(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.

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