Evaluate the Limit

Problem

(lim_x→1)((x2+7*x−8)/(x2−1))

Solution

1. Identify the indeterminate form by substituting x=1 into the expression, which results in (1+7*(1)−8)/(1−1)=0/0

2. Factor the numerator x2+7*x−8 by finding two numbers that multiply to −8 and add to 7 which are 8 and −1

x2+7*x−8=(x+8)*(x−1)

3. Factor the denominator x2−1 using the difference of squares formula a2−b2=(a−b)*(a+b)

x2−1=(x−1)*(x+1)

4. Simplify the expression by canceling the common factor (x−1) from the numerator and the denominator.

((x+8)*(x−1))/((x−1)*(x+1))=(x+8)/(x+1)

5. Evaluate the limit by substituting x=1 into the simplified expression.

(1+8)/(1+1)=9/2

Final Answer

(lim_x→1)((x2+7*x−8)/(x2−1))=9/2

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