Evaluate the Limit

Problem

(lim_x→25)((5−√(,x))/(25*x−x2))

Solution

1. Identify the indeterminate form by substituting x=25 into the expression, which results in (5−√(,25))/(25*(25)−25)=0/0

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

(5−√(,x))/(x*(25−x))

3. Apply the difference of squares formula to the term (25−x) by treating it as (5−(√(,x))2)

25−x=(5−√(,x))*(5+√(,x))

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

(lim_x→25)((5−√(,x))/(x*(5−√(,x))*(5+√(,x))))

5. Cancel the common factor (5−√(,x)) from the numerator and the denominator, provided x≠25

(lim_x→25)(1/(x*(5+√(,x))))

6. Evaluate the limit by substituting x=25 into the simplified expression.

1/(25*(5+√(,25)))

7. Simplify the numerical result.

1/(25*(5+5))=1/(25*(10))=1/250

Final Answer

(lim_x→25)((5−√(,x))/(25*x−x2))=1/250

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