Evaluate the Limit

Problem

(lim_x→∞)(√(,9*x2+x)−3*x)

Solution

1. Identify the indeterminate form. As x approaches infinity, the expression √(,9*x2+x) approaches infinity and 3*x approaches infinity, resulting in the indeterminate form ∞−∞

2. Rationalize the expression by multiplying the numerator and the denominator by the conjugate, which is √(,9*x2+x)+3*x

(√(,9*x2+x)−3*x)⋅(√(,9*x2+x)+3*x)/(√(,9*x2+x)+3*x)

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

((9*x2+x)−(3*x)2)/(√(,9*x2+x)+3*x)

(9*x2+x−9*x2)/(√(,9*x2+x)+3*x)

x/(√(,9*x2+x)+3*x)

4. Divide both the numerator and the denominator by the highest power of x in the denominator, which is x Note that for x>0 x=√(,x2)

x/x/(√(,9*x2+x)/x+(3*x)/x)

1/(√(,(9*x2+x)/(x2))+3)

1/(√(,9+1/x)+3)

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

1/(√(,9+0)+3)

1/(3+3)

1/6

Final Answer

(lim_x→∞)(√(,9*x2+x)−3*x)=1/6

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