Evaluate the Limit

Problem

(lim_x→16)((4−√(,x))/(x−16))

Solution

1. Identify the indeterminate form by substituting x=16 into the expression, which results in (4−√(,16))/(16−16)=0/0

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator, which is 4+√(,x)

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

((4−√(,x))*(4+√(,x)))/((x−16)*(4+√(,x)))=(16−x)/((x−16)*(4+√(,x)))

4. Simplify the expression by factoring out a −1 from the numerator to match the term in the denominator.

(−(x−16))/((x−16)*(4+√(,x)))=(−1)/(4+√(,x))

5. Evaluate the limit by substituting x=16 into the simplified expression.

(−1)/(4+√(,16))=(−1)/(4+4)=−1/8

Final Answer

(lim_x→16)((4−√(,x))/(x−16))=−1/8

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