Evaluate the Limit

Problem

(lim_x→9)((3−√(,x))/(9*x−x2))

Solution

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

2. Factor the denominator by taking out the greatest common factor x

(3−√(,x))/(x*(9−x))

3. Apply the difference of squares formula to the term (9−x) by treating it as (3−(√(,x))2)

(3−√(,x))/(x*(3−√(,x))*(3+√(,x)))

4. Cancel the common factor (3−√(,x)) from the numerator and the denominator, provided x≠9

1/(x*(3+√(,x)))

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

1/(9*(3+√(,9)))

6. Simplify the numerical result.

1/(9*(3+3))=1/(9*(6))=1/54

Final Answer

(lim_x→9)((3−√(,x))/(9*x−x2))=1/54

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