Evaluate the Limit

Problem

(lim_x→4)((4*x−x2)/(2−√(,x)))

Solution

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

2. Factor the numerator by taking out the common factor x

(x*(4−x))/(2−√(,x))

3. Apply the difference of squares formula to the term (4−x) to create a factor that matches the denominator.

4−x=(2−√(,x))*(2+√(,x))

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

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

5. Simplify the expression by canceling the common factor (2−√(,x)) from the numerator and denominator.

(lim_x→4)(x)*(2+√(,x))

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

4*(2+√(,4))=4*(2+2)=16

Final Answer

(lim_x→4)((4*x−x2)/(2−√(,x)))=16

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