Evaluate the Limit

Problem

(lim_x→4)((x−4)/(√(,x)−2))

Solution

1. Identify the indeterminate form by substituting x=4 into the expression, which results in (4−4)/(√(,4)−2)=0/0

2. Rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator, which is √(,x)+2

3. Expand the denominator using the difference of squares formula (a−b)*(a+b)=a2−b2 which gives (√(,x))2−2=x−4

4. Simplify the expression by canceling the common factor of x−4 from the numerator and the denominator.

5. Evaluate the limit by substituting x=4 into the remaining expression √(,x)+2

Final Answer

(lim_x→4)((x−4)/(√(,x)−2))=4

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