Evaluate the Limit

Problem

(lim_x→3)((x2−9)/(x−3))

Solution

1. Identify the indeterminate form by substituting x=3 into the expression, which results in (3−9)/(3−3)=0/0

2. Factor the numerator using the difference of squares formula, a2−b2=(a−b)*(a+b)

x2−9=(x−3)*(x+3)

3. Simplify the expression by canceling the common factor (x−3) from the numerator and the denominator, noting that x≠3

((x−3)*(x+3))/(x−3)=x+3

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

3+3=6

Final Answer

(lim_x→3)((x2−9)/(x−3))=6

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