Evaluate the Limit

Problem

(lim_x→∞)(√(,9*x2+x)−3*x)

Solution

1. Identify the indeterminate form. As x→∞ the expression √(,9*x2+x)−3*x takes the form ∞−∞

2. Rationalize the expression by multiplying and dividing by the conjugate, √(,9*x2+x)+3*x

(lim_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

(lim_x→∞)(((9*x2+x)−(3*x)2)/(√(,9*x2+x)+3*x))

(lim_x→∞)((9*x2+x−9*x2)/(√(,9*x2+x)+3*x))

(lim_x→∞)(x/(√(,9*x2+x)+3*x))

4. Divide the numerator and denominator by x Inside the square root, this is equivalent to dividing by x2 since √(,x2)=x for x>0

(lim_x→∞)(x/x/(√(,9*x2+x)/x+(3*x)/x))

(lim_x→∞)(1/(√(,(9*x2)/(x2)+x/(x2))+3))

(lim_x→∞)(1/(√(,9+1/x)+3))

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

1/(√(,9+0)+3)

1/(3+3)

Final Answer

(lim_x→∞)(√(,9*x2+x)−3*x)=1/6

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