Evaluate the Limit

Problem

(lim_x→∞)(√(,x+x2)/(5*x−x2))

Solution

1. Identify the highest power of x in the denominator to determine the growth rate, which is x2

2. Divide both the numerator and the denominator by x2 to simplify the expression for the limit at infinity.

3. Rewrite the numerator by bringing the x2 inside the square root as x4

√(,x+x2)/(x2)=√(,(x+x2)/(x4))

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

√(,1/(x3)+1/(x2))

(5*x−x2)/(x2)=5/x−1

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

(lim_x→∞)(√(,1/(x3)+1/(x2))/(5/x−1))=√(,0+0)/(0−1)

6. Calculate the final numerical value.

0/(−1)=0

Final Answer

(lim_x→∞)(√(,x+x2)/(5*x−x2))=0

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