Evaluate the Limit

Problem

(lim_x→∞)(√(,16*x2+x)−4*x)

Solution

1. Identify the indeterminate form. As x→∞ the expression √(,16*x2+x)−4*x results in the form ∞−∞

2. Multiply the expression by its conjugate over itself to rationalize the numerator.

(√(,16*x2+x)−4*x)/1⋅(√(,16*x2+x)+4*x)/(√(,16*x2+x)+4*x)

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

((16*x2+x)−(4*x)2)/(√(,16*x2+x)+4*x)

4. Combine like terms in the numerator.

(16*x2+x−16*x2)/(√(,16*x2+x)+4*x)=x/(√(,16*x2+x)+4*x)

5. Divide every term by x to evaluate the limit as x→∞ Note that for x>0 x=√(,x2)

x/x/(√(,16*x2+x)/x+(4*x)/x)

6. Simplify the expression inside the square root by dividing by x2

1/(√(,(16*x2)/(x2)+x/(x2))+4)

1/(√(,16+1/x)+4)

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

1/(√(,16+0)+4)

1/(4+4)=1/8

Final Answer

(lim_x→∞)(√(,16*x2+x)−4*x)=1/8

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