Evaluate the Limit

Problem

(lim_x→−1)((x2−8*x)/(x2−7*x−8))

Solution

1. Identify the type of limit by substituting the value x=−1 into the expression.

2. Substitute the value into the numerator: (−1)2−8*(−1)=1+8=9

3. Substitute the value into the denominator: (−1)2−7*(−1)−8=1+7−8=0

4. Analyze the result of the substitution, which is 9/0 Since the numerator is a non-zero constant and the denominator approaches zero, the limit does not exist as a finite number and represents a vertical asymptote.

5. Check the behavior from the left (x→−1. The numerator is positive (9. In the denominator, x2−7*x−8=(x−8)*(x+1) As x→−1 (x−8)→−9 and (x+1) is a small negative number. Their product is a small positive number. Thus, 9/0→∞

6. Check the behavior from the right (x→−1. The numerator is positive (9. In the denominator, (x−8)→−9 and (x+1) is a small positive number. Their product is a small negative number. Thus, 9/0→−∞

7. Conclude that because the one-sided limits do not match, the two-sided limit does not exist.

Final Answer

(lim_x→−1)((x2−8*x)/(x2−7*x−8))=Does Not Exist

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