Evaluate the Limit

Problem

(lim_x→∞)(√(,81*x2+x)−9*x)

Solution

1. Identify the indeterminate form. As x→∞ the expression √(,81*x2+x)−9*x takes the form ∞−∞

2. Multiply the expression by its conjugate over itself to rationalize the numerator.

(√(,81*x2+x)−9*x)/1⋅(√(,81*x2+x)+9*x)/(√(,81*x2+x)+9*x)

3. Simplify the numerator using the difference of squares formula (a−b)*(a+b)=a2−b2

((81*x2+x)−(9*x)2)/(√(,81*x2+x)+9*x)

4. Combine like terms in the numerator.

(81*x2+x−81*x2)/(√(,81*x2+x)+9*x)=x/(√(,81*x2+x)+9*x)

5. Divide both the numerator and the denominator by x Inside the square root, this is equivalent to dividing by x2

x/x/(√(,81*x2+x)/x+(9*x)/x)

6. Simplify the expression inside the square root.

1/(√(,(81*x2)/(x2)+x/(x2))+9)

1/(√(,81+1/x)+9)

7. Evaluate the limit as x→∞ The term 1/x approaches 0

1/(√(,81+0)+9)

1/(9+9)

Final Answer

(lim_x→∞)(√(,81*x2+x)−9*x)=1/18

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