Evaluate the Limit

Problem

(lim_x→−∞)(x+√(,x2+2*x))

Solution

1. Identify the indeterminate form. As x→−∞ the expression x+√(,x2+2*x) results in the form −∞+∞

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

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

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

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

(lim_x→−∞)((−2*x)/(x−√(,x2+2*x)))

4. Factor x2 out of the square root in the denominator. Note that √(,x2)=|x| and since x→−∞ |x|=−x

(lim_x→−∞)((−2*x)/(x−√(,x2*(1+2/x))))

(lim_x→−∞)((−2*x)/(x−(−x)√(,1+2/x)))

5. Simplify the denominator by combining the x terms.

(lim_x→−∞)((−2*x)/(x+x√(,1+2/x)))

(lim_x→−∞)((−2*x)/(x*(1+√(,1+2/x))))

6. Divide the numerator and denominator by x

(lim_x→−∞)((−2)/(1+√(,1+2/x)))

7. Evaluate the limit as x→−∞ The term 2/x approaches 0

(−2)/(1+√(,1+0))

(−2)/(1+1)

Final Answer

(lim_x→−∞)(x+√(,x2+2*x))=−1

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