Evaluate the Limit

Problem

(lim_x→∞)(√(,49*x2+x)−7*x)

Solution

1. Identify the indeterminate form. As x→∞ the expression √(,49*x2+x)−7*x takes the form ∞−∞

2. Rationalize the expression by multiplying the numerator and the denominator by the conjugate, √(,49*x2+x)+7*x

(lim_x→∞)(((√(,49*x2+x)−7*x)*(√(,49*x2+x)+7*x))/(√(,49*x2+x)+7*x))

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

(lim_x→∞)(((49*x2+x)−(7*x)2)/(√(,49*x2+x)+7*x))

(lim_x→∞)((49*x2+x−49*x2)/(√(,49*x2+x)+7*x))

(lim_x→∞)(x/(√(,49*x2+x)+7*x))

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

(lim_x→∞)(x/x/(√(,49*x2+x)/x+(7*x)/x))

(lim_x→∞)(1/(√(,(49*x2)/(x2)+x/(x2))+7))

(lim_x→∞)(1/(√(,49+1/x)+7))

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

1/(√(,49+0)+7)

1/(7+7)

Final Answer

(lim_x→∞)(√(,49*x2+x)−7*x)=1/14

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