Evaluate the Limit

Problem

(lim_x→−2)((2−|x|)/(2+x))

Solution

1. Analyze the behavior of the absolute value function |x| near x=−2 Since we are approaching −2 which is a negative value, we use the definition |x|=−x for x<0

2. Substitute the expression for the absolute value into the limit.

(lim_x→−2)((2−(−x))/(2+x))

3. Simplify the numerator by distributing the negative sign.

(lim_x→−2)((2+x)/(2+x))

4. Divide the common terms in the numerator and the denominator, noting that x≠−2 as we approach the limit.

(lim_x→−2)(1)

5. Evaluate the limit of the constant.

1

Final Answer

(lim_x→−2)((2−|x|)/(2+x))=1

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