Evaluate the Limit

Problem

(lim_x→8)((x−8)/(x2−7*x−8))

Solution

1. Identify the indeterminate form by substituting x=8 into the expression, which results in (8−8)/(64−56−8)=0/0

2. Factor the quadratic expression in the denominator by finding two numbers that multiply to −8 and add to −7

3. Rewrite the denominator using the factors (x−8) and (x+1)

x2−7*x−8=(x−8)*(x+1)

4. Simplify the expression by canceling the common factor (x−8) from the numerator and the denominator, noting that x≠8 as we approach the limit.

(x−8)/((x−8)*(x+1))=1/(x+1)

5. Evaluate the limit by substituting x=8 into the simplified expression.

1/(8+1)=1/9

Final Answer

(lim_x→8)((x−8)/(x2−7*x−8))=1/9

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