Evaluate the Limit

Problem

(lim_x→64)((√(,x)−8)/(x−64))

Solution

1. Identify the indeterminate form by substituting x=64 into the expression, which results in (√(,64)−8)/(64−64)=0/0

2. Factor the denominator using the difference of squares formula, a2−b2=(a−b)*(a+b) where x=(√(,x))2 and 64=8

x−64=(√(,x)−8)*(√(,x)+8)

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

(lim_x→64)((√(,x)−8)/((√(,x)−8)*(√(,x)+8)))

4. Cancel the common factor (√(,x)−8) from the numerator and the denominator.

(lim_x→64)(1/(√(,x)+8))

5. Evaluate the limit by substituting x=64 into the simplified expression.

1/(√(,64)+8)=1/(8+8)=1/16

Final Answer

(lim_x→64)((√(,x)−8)/(x−64))=1/16

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