Evaluate the Limit

Problem

(lim_x→∞)(√(,36*x2+x))−6*x

Solution

1. Identify the indeterminate form. As x approaches ∞ the expression √(,36*x2+x) approaches ∞ and 6*x approaches ∞ resulting in the form ∞−∞

2. Rationalize the expression by multiplying the numerator and the denominator by the conjugate, √(,36*x2+x)+6*x

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

(lim_x→∞)(((√(,36*x2+x)−6*x)*(√(,36*x2+x)+6*x))/(√(,36*x2+x)+6*x))

(lim_x→∞)(((36*x2+x)−(6*x)2)/(√(,36*x2+x)+6*x))

(lim_x→∞)((36*x2+x−36*x2)/(√(,36*x2+x)+6*x))

(lim_x→∞)(x/(√(,36*x2+x)+6*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)

(lim_x→∞)(x/x/(√(,36*x2+x)/x+(6*x)/x))

(lim_x→∞)(1/(√(,(36*x2+x)/(x2))+6))

(lim_x→∞)(1/(√(,36+1/x)+6))

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

1/(√(,36+0)+6)

1/(6+6)

1/12

Final Answer

(lim_x→∞)(√(,36*x2+x))−6*x=1/12

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