Evaluate the Limit

Problem

(lim_x→2)(√(,(2*x3+4)/(x2+1)))

Solution

1. Apply the limit property for continuous functions, which allows moving the limit inside the square root.

√(,(lim_x→2)((2*x3+4)/(x2+1)))

2. Apply the quotient rule for limits, provided the denominator is non-zero at the limit point.

√(,(lim_x→2)(2*x3+4)/(lim_x→2)(x2+1))

3. Substitute the value x=2 into the expression since the function is defined and continuous at that point.

√(,(2*(2)3+4)/((2)2+1))

4. Simplify the numerator and the denominator.

√(,(2*(8)+4)/(4+1))

5. Calculate the final values inside the square root.

√(,(16+4)/5)

√(,20/5)

6. Divide the terms and simplify the radical.

√(,4)

2

Final Answer

(lim_x→2)(√(,(2*x3+4)/(x2+1)))=2

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