Evaluate the Limit

Problem

(lim_x→5)((x2−5*x)/(x2−4*x−5))

Solution

1. Substitute the value x=5 into the expression to check for an indeterminate form.

(5−5*(5))/(5−4*(5)−5)=(25−25)/(25−20−5)=0/0

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

x2−5*x=x*(x−5)

3. Factor the quadratic trinomial in the denominator by finding two numbers that multiply to −5 and add to −4

x2−4*x−5=(x−5)*(x+1)

4. Simplify the expression by canceling the common factor (x−5) from the numerator and the denominator.

(x*(x−5))/((x−5)*(x+1))=x/(x+1)

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

(lim_x→5)(x/(x+1))=5/(5+1)=5/6

Final Answer

(lim_x→5)((x2−5*x)/(x2−4*x−5))=5/6

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