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 of the numerator, which is √(,x)+2

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

((√(,x)−2)*(√(,x)+2))/((x−4)*(√(,x)+2))=(x−4)/((x−4)*(√(,x)+2))

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

1/(√(,x)+2)

5. Evaluate the limit by substituting x=4 into the simplified expression.

1/(√(,4)+2)=1/(2+2)=1/4

Final Answer

(lim_x→4)((√(,x)−2)/(x−4))=1/4

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