Evaluate the Limit

Problem

(lim_x→2)((x2−2*x)/(x2−x−2))

Solution

1. Substitute the value x=2 into the expression to check for an indeterminate form.

(2−2*(2))/(2−2−2)=0/0

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

x2−2*x=x*(x−2)

3. Factor the quadratic trinomial in the denominator.

x2−x−2=(x−2)*(x+1)

4. Simplify the expression by canceling the common factor (x−2) from the numerator and the denominator.

(x*(x−2))/((x−2)*(x+1))=x/(x+1)

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

(lim_x→2)(x/(x+1))=2/(2+1)

6. Calculate the final numerical value.

2/(2+1)=2/3

Final Answer

(lim_x→2)((x2−2*x)/(x2−x−2))=2/3

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