Evaluate the Limit

Problem

(lim_x→16)((√(,x)−4)/(x−16))

Solution

1. Identify the form of the limit by substituting x=16 into the expression.

(√(,16)−4)/(16−16)=0/0

2. Factor the denominator using the difference of squares formula, a2−b2=(a−b)*(a+b) where x=(√(,x))2 and 16=4

x−16=(√(,x)−4)*(√(,x)+4)

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

(lim_x→16)((√(,x)−4)/((√(,x)−4)*(√(,x)+4)))

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

(lim_x→16)(1/(√(,x)+4))

5. Evaluate the limit by substituting x=16 into the simplified expression.

1/(√(,16)+4)=1/(4+4)

Final Answer

(lim_x→16)((√(,x)−4)/(x−16))=1/8

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