Evaluate the Limit

Problem

(lim_x→−4)((4−|x|)/(4+x))

Solution

1. Analyze the absolute value expression |x| as x approaches −4 Since we are looking at values of x near −4 x is negative.

2. Apply the definition of absolute value for negative numbers, where |x|=−x when x<0

3. Substitute −x for |x| in the limit expression.

(lim_x→−4)((4−(−x))/(4+x))

4. Simplify the numerator by distributing the negative sign.

(lim_x→−4)((4+x)/(4+x))

5. Divide the common terms in the numerator and denominator, noting that as x approaches −4 the expression (4+x)/(4+x) equals 1 for all x≠−4

(lim_x→−4)(1)

6. Evaluate the limit of the constant.

1

Final Answer

(lim_x→−4)((4−|x|)/(4+x))=1

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