Evaluate the Limit

Problem

(lim_x→7)((x−7)/(x2−6*x−7))

Solution

1. Identify the form of the limit by substituting x=7 into the expression.

(7−7)/(7−6*(7)−7)=0/(49−42−7)=0/0

2. Factor the quadratic expression in the denominator by finding two numbers that multiply to −7 and add to −6

x2−6*x−7=(x−7)*(x+1)

3. Substitute the factored form back into the limit expression.

(lim_x→7)((x−7)/((x−7)*(x+1)))

4. Simplify the expression by canceling the common factor (x−7) from the numerator and the denominator, noting that x≠7

(lim_x→7)(1/(x+1))

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

1/(7+1)=1/8

Final Answer

(lim_x→7)((x−7)/(x2−6*x−7))=1/8

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