Evaluate the Limit

Problem

(lim_x→−9)((9−|x|)/(9+x))

Solution

1. Identify the behavior of the absolute value function |x| as x approaches −9 Since we are looking at values of x near −9 x is negative.

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

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

(9−(−x))/(9+x)

4. Simplify the numerator by applying the double negative.

(9+x)/(9+x)

5. Divide the common terms in the numerator and denominator, noting that for x≠−9 the expression equals 1

(9+x)/(9+x)=1

6. Evaluate the limit of the constant value.

(lim_x→−9)(1)=1

Final Answer

(lim_x→−9)((9−|x|)/(9+x))=1

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