Let h(x)= |x-4|(x^ 2 -16) x-4 (x 4)

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

Problem

Let h(x)=∣x−4∣(x2−16)x−4(x≠4),h(x)=\frac{|x-4|(x^{2}-16)}{x-4}\quad (x\neq4), and define h(4)=kh(4)=k, where kk is a constant. Use a combination of algebraic techniques and limit properties to answer the following. (1) Compute lim⁡x→4−h(x)\displaystyle\lim_{x\to4^-}h(x) and lim⁡x→4+h(x)\displaystyle\lim_{x\to4^+}h(x) separately, showing each algebraic step. (2) Determine whether the one-sided limits you found in part (a) are equal. If they are equal, state their common value; if not, explain why they differ. (3) Find the value of kk that makes hh continuous at x=4x=4. Justify your answer using your results from parts (a) and (b). (4) Provide a brief written justification that each algebraic step you used in parts (a) and (b) is valid for evaluating a limit, citing appropriate limit properties.

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