Evaluate the Limit

Problem

(lim_x→3)((x−3)/(x2−9))

Solution

1. Identify the form of the limit by substituting x=3 into the expression, which results in the indeterminate form 0/0

2. Factor the denominator 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 as it approaches the limit.

(x−3)/((x−3)*(x+3))=1/(x+3)

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

(lim_x→3)(1/(x+3))=1/(3+3)

5. Calculate the final numerical value.

1/(3+3)=1/6

Final Answer

(lim_x→3)((x−3)/(x2−9))=1/6

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