Evaluate the Limit

Problem

(lim_x→−5)((5−|x|)/(5+x))

Solution

1. Analyze the absolute value expression |x| as x approaches −5 Since we are looking at values of x near −5 x is negative.

2. Apply the definition of absolute value for x<0 which states |x|=−x

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

(lim_x→−5)((5−(−x))/(5+x))

4. Simplify the numerator by distributing the negative sign.

(lim_x→−5)((5+x)/(5+x))

5. Reduce the fraction by dividing the numerator and denominator by the common factor (5+x) noting that x≠−5 as we approach the limit.

(lim_x→−5)(1)

6. Evaluate the limit of the constant.

1

Final Answer

(lim_x→−5)((5−|x|)/(5+x))=1

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