Evaluate the Limit

Problem

(lim_x→2)(√(,(5*x2+5)/(5*x−1)))

Solution

1. Apply the limit property for roots, which allows the limit to be moved inside the square root function.

√(,(lim_x→2)((5*x2+5)/(5*x−1)))

2. Apply the limit property for quotients, moving the limit into the numerator and the denominator.

√(,(lim_x→2)(5*x2+5)/(lim_x→2)(5*x−1))

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

√(,(5*(2)2+5)/(5*(2)−1))

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

√(,(5*(4)+5)/(10−1))

5. Calculate the final values inside the square root.

√(,(20+5)/9)

6. Simplify the fraction and take the square root.

√(,25/9)

7. Evaluate the square root of the fraction.

5/3

Final Answer

(lim_x→2)(√(,(5*x2+5)/(5*x−1)))=5/3

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