Evaluate the Limit

Problem

(lim_x→9)((9−x)/(√(,x)−3))

Solution

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

2. Factor the numerator using the difference of squares formula, a2−b2=(a−b)*(a+b) where 9−x=(3−√(,x))*(3+√(,x))

3. Rewrite the numerator to match the denominator by factoring out a negative sign: (3−√(,x))=−(√(,x)−3)

4. Simplify the expression by canceling the common factor (√(,x)−3) from the numerator and the denominator.

5. Evaluate the limit by substituting x=9 into the remaining simplified expression.

Final Answer

(lim_x→9)((9−x)/(√(,x)−3))=−6

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