Mathematics

Problem

Let

h(x)=\frac{|x-4|(x^{2}-16)}{x-4}\quad (x\neq4),

and define

h(4)=k

, where

k

is a constant. Use a combination of algebraic techniques and limit properties to answer the following. a. Compute

\displaystyle\lim_{x\to4^-}h(x)

and

\displaystyle\lim_{x\to4^+}h(x)

separately, showing each algebraic step. b. 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. c. Find the value of

k

that makes

h

continuous at

x=4

. Justify your answer using your results from parts (a) and (b). d. 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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