Evaluate the Limit

Problem

(lim_x→3)((x2−x−6)/(x−3))

Solution

1. Identify the indeterminate form by substituting x=3 into the expression, which results in (3−3−6)/(3−3)=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. Simplify the expression by canceling the common factor (x−3) from the numerator and the denominator.

((x−3)*(x+2))/(x−3)=x+2

4. Evaluate the limit by substituting x=3 into the simplified expression.

(lim_x→3)(x+2)=3+2=5

Final Answer

(lim_x→3)((x2−x−6)/(x−3))=5

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