Evaluate the Limit

Problem

(lim_x→4)(1/(√(,x)−2)−4/(x−4))

Solution

1. Factor the denominator of the second term using the difference of squares formula, noting that x−4=(√(,x)−2)*(√(,x)+2)

1/(√(,x)−2)−4/((√(,x)−2)*(√(,x)+2))

2. Find a common denominator to combine the fractions by multiplying the first term by (√(,x)+2)/(√(,x)+2)

(√(,x)+2)/((√(,x)−2)*(√(,x)+2))−4/((√(,x)−2)*(√(,x)+2))

3. Combine the numerators over the common denominator.

(√(,x)+2−4)/((√(,x)−2)*(√(,x)+2))

4. Simplify the numerator by performing the subtraction.

(√(,x)−2)/((√(,x)−2)*(√(,x)+2))

5. Cancel the common factor of √(,x)−2 from the numerator and the denominator, provided x≠4

1/(√(,x)+2)

6. Evaluate the limit by substituting x=4 into the simplified expression.

1/(√(,4)+2)

1/(2+2)

1/4

Final Answer

(lim_x→4)(1/(√(,x)−2)−4/(x−4))=1/4

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