Evaluate the Limit

Problem

(lim_x→4)((√(,x)−2)/(x−4))

Solution

1. Identify the indeterminate form by substituting x=4 into the expression, which results in (√(,4)−2)/(4−4)=0/0

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate expression √(,x)+2

3. Simplify the numerator using the difference of squares formula (a−b)*(a+b)=a2−b2 which gives (√(,x))2−2=x−4

4. Cancel the common factor of x−4 from the numerator and the denominator, provided x≠4

5. Evaluate the limit by substituting x=4 into the remaining simplified expression 1/(√(,x)+2)

Final Answer

(lim_x→4)((√(,x)−2)/(x−4))=1/4

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