Evaluate the Limit

Problem

(lim_x→−6)((6−|x|)/(6+x))

Solution

1. Analyze the behavior of the absolute value function |x| as x approaches −6 Since we are looking at values of x near −6 x is negative.

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

3. Rewrite the limit expression by replacing |x| with −x

(lim_x→−6)((6−(−x))/(6+x))

4. Simplify the numerator by applying the double negative.

(lim_x→−6)((6+x)/(6+x))

5. Reduce the fraction, noting that for all x≠−6 the expression equals 1

(lim_x→−6)(1)

6. Evaluate the limit of the constant.

1

Final Answer

(lim_x→−6)((6−|x|)/(6+x))=1

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