Evaluate the Limit

Problem

(lim_x→7)((x2−5*x−14)/(x−7))

Solution

1. Identify the indeterminate form by substituting x=7 into the expression, which results in 0/0

2. Factor the numerator x2−5*x−14 by finding two numbers that multiply to −14 and add to −5

3. Rewrite the quadratic expression as (x−7)*(x+2)

4. Cancel the common factor of (x−7) from the numerator and the denominator, noting that x≠7

5. Evaluate the limit by substituting x=7 into the remaining simplified expression x+2

Final Answer

(lim_x→7)((x2−5*x−14)/(x−7))=9

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