Evaluate the Limit

Problem

(lim_x→9)((x−9)/(√(,x)−3))

Solution

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

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

3. Rewrite the expression with the factored numerator:

((√(,x)−3)*(√(,x)+3))/(√(,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:

√(,9)+3

6. Calculate the final value:

3+3=6

Final Answer

(lim_x→9)((x−9)/(√(,x)−3))=6

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