Evaluate the Limit

Problem

(lim_x→2)((x2+4*x−12)/(x2−4))

Solution

1. Identify the indeterminate form by substituting x=2 into the expression, which results in 0/0

2. Factor the numerator x2+4*x−12 into (x+6)*(x−2)

3. Factor the denominator x2−4 as a difference of squares into (x+2)*(x−2)

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

5. Evaluate the limit by substituting x=2 into the remaining expression (x+6)/(x+2)

Final Answer

(lim_x→2)((x2+4*x−12)/(x2−4))=2

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