Evaluate the Limit

Problem

(lim_x→∞)(√(,x+x^2)/(5*x−x^2))

Solution

1. Identify the highest power of x in the denominator to determine the growth rate, which is x^2

2. Divide both the numerator and the denominator by x^2 to simplify the expression for the limit at infinity.

3. Rewrite the numerator by bringing the x^2 inside the square root as x^4

√(,x+x^2)/(x^2)=√(,(x+x^2)/(x^4))

4. Simplify the terms inside the square root and the terms in the denominator.

√(,1/(x^3)+1/(x^2))

(5*x−x^2)/(x^2)=5/x−1

5. Evaluate the limit as x→∞ by noting that any term with x in the denominator approaches 0

(lim_x→∞)(√(,1/(x^3)+1/(x^2))/(5/x−1))=√(,0+0)/(0−1)

6. Calculate the final numerical value.

0/(−1)=0

Final Answer

(lim_x→∞)(√(,x+x^2)/(5*x−x^2))=0

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