Evaluate the Limit

Problem

(lim_x→36)((6−√(,x))/(36*x−x2))

Solution

1. Identify the indeterminate form by substituting x=36 into the expression, which results in (6−√(,36))/(36*(36)−36)=0/0

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

(6−√(,x))/(x*(36−x))

3. Apply the difference of squares formula to the term (36−x) by treating it as (6−(√(,x))2)

36−x=(6−√(,x))*(6+√(,x))

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

(lim_x→36)((6−√(,x))/(x*(6−√(,x))*(6+√(,x))))

5. Cancel the common factor (6−√(,x)) from the numerator and the denominator, provided x≠36

(lim_x→36)(1/(x*(6+√(,x))))

6. Evaluate the limit by substituting x=36 into the simplified expression.

1/(36*(6+√(,36)))

7. Simplify the numerical result.

1/(36*(6+6))=1/(36*(12))=1/432

Final Answer

(lim_x→36)((6−√(,x))/(36*x−x2))=1/432

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