Evaluate the Limit

Problem

(lim_x→2)((x2+2*x−8)/(x2−4))

Solution

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

(2+2*(2)−8)/(2−4)=0/0

2. Factor the numerator by finding two numbers that multiply to −8 and add to 2

x2+2*x−8=(x+4)*(x−2)

3. Factor the denominator using the difference of squares formula a2−b2=(a−b)*(a+b)

x2−4=(x−2)*(x+2)

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

((x+4)*(x−2))/((x−2)*(x+2))=(x+4)/(x+2)

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

(2+4)/(2+2)=6/4

6. Reduce the fraction to its simplest form.

6/4=3/2

Final Answer

(lim_x→2)((x2+2*x−8)/(x2−4))=3/2

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