Evaluate the Limit

Problem

(lim_x→2)(√(,(5*x2+5)/(9*x−2)))

Solution

1. Apply the limit property for roots, which allows moving the limit inside the square root function.

√(,(lim_x→2)((5*x2+5)/(9*x−2)))

2. Apply the quotient rule for limits by taking the limit of the numerator and the denominator separately.

√(,(lim_x→2)(5*x2+5)/(lim_x→2)(9*x−2))

3. Substitute the value x=2 into the expression, as the function is continuous at this point.

√(,(5*(2)2+5)/(9*(2)−2))

4. Simplify the numerator and denominator by performing the arithmetic operations.

√(,(5*(4)+5)/(18−2))

√(,(20+5)/16)

√(,25/16)

5. Evaluate the square root of the resulting fraction.

5/4

Final Answer

(lim_x→2)(√(,(5*x2+5)/(9*x−2)))=5/4

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