Evaluate the Limit

Problem

(lim_x→−2)((x2−x−6)/(x+2))

Solution

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

2. Factor the quadratic expression in the numerator by finding two numbers that multiply to −6 and add to −1

x2−x−6=(x−3)*(x+2)

3. Substitute the factored form back into the limit expression.

(lim_x→−2)(((x−3)*(x+2))/(x+2))

4. Simplify the expression by canceling the common factor (x+2) from the numerator and the denominator, noting that x≠−2

(lim_x→−2)(x−3)

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

−2−3=−5

Final Answer

(lim_x→−2)((x2−x−6)/(x+2))=−5

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