Evaluate the Limit

Problem

(lim_x→16)((4−√(,x))/(16*x−x2))

Solution

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

2. Factor the denominator by taking out the greatest common factor x

(4−√(,x))/(x*(16−x))

3. Apply the difference of squares formula to the term (16−x) by treating it as (4−(√(,x))2)

16−x=(4−√(,x))*(4+√(,x))

4. Substitute the factored form back into the limit expression.

(lim_x→16)((4−√(,x))/(x*(4−√(,x))*(4+√(,x))))

5. Simplify the expression by canceling the common factor (4−√(,x)) from the numerator and the denominator.

(lim_x→16)(1/(x*(4+√(,x))))

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

1/(16*(4+√(,16)))

1/(16*(4+4))

1/(16*(8))

1/128

Final Answer

(lim_x→16)((4−√(,x))/(16*x−x2))=1/128

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