Evaluate the Limit

Problem

(lim_x→2)((x2−4*x+4)/(x2+x−6))

Solution

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

(2−4*(2)+4)/(2+2−6)=0/0

2. Factor the numerator, which is a perfect square trinomial.

x2−4*x+4=(x−2)2

3. Factor the denominator by finding two numbers that multiply to −6 and add to 1

x2+x−6=(x−2)*(x+3)

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

((x−2)2)/((x−2)*(x+3))=(x−2)/(x+3)

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

(2−2)/(2+3)=0/5=0

Final Answer

(lim_x→2)((x2−4*x+4)/(x2+x−6))=0

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