MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
The function g(x)=x+1x2+2x+1−(x+1) is defined for all real x=−1.
Consider the limit of g(x) as x approaches −1.
(1) Part 1: (a) Identify the indeterminate form that arises when direct substitution x=−1 is attempted.
(2) Part 2: (b) Provide a verbal explanation, without carrying out any algebra, of why applying only a single factoring step or only a single multiplication by a conjugate is insufficient to resolve this indeterminate form.
(3) Part 3: (c) Describe a complete algebraic strategy—listing the main steps in order—that will transform g(x) into an expression for which direct substitution at x=−1 becomes possible. Justify why each step you list is necessary.