Evaluate the Limit

Problem

(lim_x→3)((√(,x+1)−2)/(x−3))

Solution

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

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

(√(,x+1)−2)*(√(,x+1)+2)=(x+1)−4

4. Simplify the numerator to x−3

(x−3)/((x−3)*(√(,x+1)+2))

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

1/(√(,x+1)+2)

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

1/(√(,3+1)+2)=1/(2+2)

Final Answer

(lim_x→3)((√(,x+1)−2)/(x−3))=1/4

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