Evaluate the Limit

Problem

(lim_x→36)((√(,x)−6)/(x−36))

Solution

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

2. Factor the denominator using the difference of squares formula, a2−b2=(a−b)*(a+b) where x=(√(,x))2 and 36=6

x−36=(√(,x)−6)*(√(,x)+6)

3. Substitute the factored form into the limit expression.

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

4. Simplify the expression by canceling the common factor (√(,x)−6) from the numerator and the denominator.

(lim_x→36)(1/(√(,x)+6))

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

1/(√(,36)+6)=1/(6+6)

Final Answer

(lim_x→36)((√(,x)−6)/(x−36))=1/12

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