Evaluate the Limit

Problem

(lim_x→49)((7−√(,x))/(49*x−x2))

Solution

1. Identify the indeterminate form by substituting x=49 into the expression, which results in (7−√(,49))/(49*(49)−49)=0/0

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

(7−√(,x))/(x*(49−x))

3. Apply the difference of squares formula to the term (49−x) by treating it as (7−(√(,x))2)

49−x=(7−√(,x))*(7+√(,x))

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

(lim_x→49)((7−√(,x))/(x*(7−√(,x))*(7+√(,x))))

5. Cancel the common factor (7−√(,x)) from the numerator and the denominator, provided x≠49

(lim_x→49)(1/(x*(7+√(,x))))

6. Evaluate the limit by substituting x=49 into the simplified expression.

1/(49*(7+√(,49)))

7. Simplify the numerical result.

1/(49*(7+7))=1/(49*(14))=1/686

Final Answer

(lim_x→49)((7−√(,x))/(49*x−x2))=1/686

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