Consider the composite expression Q(x)= x^ 2 +6x+9 x+3 - (x+3) x

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

Problem

Consider the composite expression Q(x)=x2+6x+9x+3−(x+3)x.Q(x)=\dfrac{\dfrac{x^{2}+6x+9}{x+3} - (x+3)}{x}. Assume x≠0x \neq 0 and x≠−3x \neq -3. (a) Determine lim⁡x→0Q(x)\displaystyle \lim_{x\to 0} Q(x). Show your work in a clear sequence of algebraic steps. (b) Verify that each intermediate form encountered in your work is indeterminate, and explain how your algebraic manipulations resolve these indeterminate forms. (c) Suppose the constant term 99 in the numerator of Q(x)Q(x) is replaced by a real parameter bb, producing Rb(x)=x2+6x+bx+3−(x+3)x.R_b(x)=\dfrac{\dfrac{x^{2}+6x+b}{x+3} - (x+3)}{x}. Find all real values of bb for which lim⁡x→0Rb(x)\displaystyle \lim_{x\to 0} R_b(x) does not exist, and justify your answer.

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