Evaluate the Limit

Problem

(lim_x→4)((2−√(,x))/(4*x−x2))

Solution

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

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

(2−√(,x))/(x*(4−x))

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

4−x=(2−√(,x))*(2+√(,x))

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

(lim_x→4)((2−√(,x))/(x*(2−√(,x))*(2+√(,x))))

5. Cancel the common factor (2−√(,x)) from the numerator and the denominator, provided x≠4

(lim_x→4)(1/(x*(2+√(,x))))

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

1/(4*(2+√(,4)))

7. Simplify the numerical result.

1/(4*(2+2))=1/16

Final Answer

(lim_x→4)((2−√(,x))/(4*x−x2))=1/16

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