Evaluate the Limit

Problem

(lim_x→4)((√(,x+5)−3)/(x−4))

Solution

1. Identify the indeterminate form by substituting x=4 into the expression, which results in (√(,9)−3)/(4−4)=0/0

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate expression √(,x+5)+3

3. Expand the numerator using the difference of squares formula (a−b)*(a+b)=a2−b2

(√(,x+5)−3)*(√(,x+5)+3)=(x+5)−9

4. Simplify the numerator to obtain x−4

(x−4)/((x−4)*(√(,x+5)+3))

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

1/(√(,x+5)+3)

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

1/(√(,4+5)+3)=1/(3+3)

Final Answer

(lim_x→4)((√(,x+5)−3)/(x−4))=1/6

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