Evaluate the Limit

Problem

(lim_x→9)((√(,x)−3)/(x−9))

Solution

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

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator, which is √(,x)+3

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

4. Simplify the expression by canceling the common factor (x−9) from the numerator and the denominator.

5. Evaluate the limit by substituting x=9 into the remaining simplified expression 1/(√(,x)+3)

Final Answer

(lim_x→9)((√(,x)−3)/(x−9))=1/6

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