Evaluate the Limit

Problem

(lim_x→7)((√(,x+2)−3)/(x−7))

Solution

1. Identify the indeterminate form by substituting x=7 into the expression, which results in (√(,7+2)−3)/(7−7)=0/0

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

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

((√(,x+2)−3)*(√(,x+2)+3))/((x−7)*(√(,x+2)+3))=((x+2)−9)/((x−7)*(√(,x+2)+3))

4. Simplify the numerator to find a common factor with the denominator.

(x−7)/((x−7)*(√(,x+2)+3))

5. Cancel the common factor (x−7) from the numerator and the denominator, assuming x≠7

1/(√(,x+2)+3)

6. Evaluate the limit by substituting x=7 into the simplified expression.

1/(√(,7+2)+3)=1/(3+3)

Final Answer

(lim_x→7)((√(,x+2)−3)/(x−7))=1/6

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