Evaluate the Limit

Problem

(lim_x→∞)(√(,x+x^2)/(2*x−x^2))

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 x^2 which is the dominant term in the denominator.

3. Simplify the expression by moving x^2 inside the square root in the numerator as x^4

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

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

(lim_x→∞)(√(,x/(x^4)+(x^2)/(x^4))/((2*x)/(x^2)−(x^2)/(x^2)))

5. Reduce the fractions within the limit expression.

(lim_x→∞)(√(,1/(x^3)+1/(x^2))/(2/x−1))

6. Apply the limit laws, noting that 1/(x^n)→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+x^2)/(2*x−x^2))=0

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