Evaluate the Limit

Problem

(lim_x→3)((√(,x+6)−3)/(x−3))

Solution

1. Identify the indeterminate form by substituting x=3 into the expression, which results in (√(,3+6)−3)/(3−3)=0/0

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator, which is √(,x+6)+3

3. Expand the numerator using the difference of squares formula (a−b)*(a+b)=a2−b2

(lim_x→3)(((√(,x+6)−3)*(√(,x+6)+3))/((x−3)*(√(,x+6)+3)))

(lim_x→3)(((x+6)−9)/((x−3)*(√(,x+6)+3)))

4. Simplify the numerator to x−3

(lim_x→3)((x−3)/((x−3)*(√(,x+6)+3)))

5. Cancel the common factor of x−3 from the numerator and the denominator.

(lim_x→3)(1/(√(,x+6)+3))

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

1/(√(,3+6)+3)=1/(3+3)=1/6

Final Answer

(lim_x→3)((√(,x+6)−3)/(x−3))=1/6

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