Evaluate the Limit

Problem

(lim_x→∞)(√(,25*x2+x)−5*x)

Solution

1. Identify the indeterminate form. As x→∞ the expression √(,25*x2+x)−5*x takes the form ∞−∞

2. Rationalize the expression by multiplying and dividing by the conjugate, √(,25*x2+x)+5*x

(√(,25*x2+x)−5*x)⋅(√(,25*x2+x)+5*x)/(√(,25*x2+x)+5*x)

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

((25*x2+x)−(5*x)2)/(√(,25*x2+x)+5*x)

(25*x2+x−25*x2)/(√(,25*x2+x)+5*x)

x/(√(,25*x2+x)+5*x)

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

x/x/(√(,25*x2+x)/x+(5*x)/x)

1/(√(,(25*x2+x)/(x2))+5)

1/(√(,25+1/x)+5)

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

1/(√(,25+0)+5)

1/(5+5)

1/10

Final Answer

(lim_x→∞)(√(,25*x2+x)−5*x)=1/10

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