Evaluate the Limit

Problem

(lim_x→∞)(√(,x+x2)/(2*x−x2))

Solution

1. Identify the highest power of x in the denominator to determine the growth rate of the expression as x approaches infinity.

2. Divide both the numerator and the denominator by x2 which is the dominant term in the denominator.

3. Simplify the expression by moving x2 inside the square root in the numerator as x4

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

4. Distribute the division across the terms inside and outside the square root.

(lim_x→∞)(√(,x/(x4)+(x2)/(x4))/((2*x)/(x2)−(x2)/(x2)))

5. Reduce the fractions within the limit expression.

(lim_x→∞)(√(,1/(x3)+1/(x2))/(2/x−1))

6. Apply the limit laws, noting that 1/(xn)→0 as x→∞ for any n>0

√(,0+0)/(0−1)

7. Calculate the final numerical value.

0/(−1)=0

Final Answer

(lim_x→∞)(√(,x+x2)/(2*x−x2))=0

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