Evaluate the Limit

Problem

(lim_x→7)((x2−7*x)/(x2−6*x−7))

Solution

1. Substitute the value x=7 into the expression to check for an indeterminate form.

(7−7*(7))/(7−6*(7)−7)=(49−49)/(49−42−7)=0/0

2. Factor the numerator by taking out the greatest common factor x

x2−7*x=x*(x−7)

3. 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)

4. Simplify the rational expression by canceling the common factor (x−7) from the numerator and the denominator.

(x*(x−7))/((x−7)*(x+1))=x/(x+1)

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

(lim_x→7)(x/(x+1))=7/(7+1)

6. Calculate the final numerical value.

7/(7+1)=7/8

Final Answer

(lim_x→7)((x2−7*x)/(x2−6*x−7))=7/8

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