Evaluate the Limit

Problem

(lim_x→−∞)(x+√(,x2+8*x))

Solution

1. Identify the indeterminate form. As x→−∞ the expression x+√(,x2+8*x) results in the form −∞+∞

2. Rationalize the expression by multiplying and dividing by the conjugate, x−√(,x2+8*x)

L=(lim_x→−∞)(((x+√(,x2+8*x))*(x−√(,x2+8*x)))/(x−√(,x2+8*x)))

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

L=(lim_x→−∞)((x2−(x2+8*x))/(x−√(,x2+8*x)))

L=(lim_x→−∞)((−8*x)/(x−√(,x2+8*x)))

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

L=(lim_x→−∞)((−8*x)/(x−√(,x2*(1+8/x))))

L=(lim_x→−∞)((−8*x)/(x−(−x)√(,1+8/x)))

5. Simplify the denominator by combining the x terms.

L=(lim_x→−∞)((−8*x)/(x+x√(,1+8/x)))

L=(lim_x→−∞)((−8*x)/(x*(1+√(,1+8/x))))

6. Divide the numerator and denominator by x and evaluate the limit as x→−∞

L=(lim_x→−∞)((−8)/(1+√(,1+8/x)))

L=(−8)/(1+√(,1+0))

L=(−8)/2

Final Answer

(lim_x→−∞)(x+√(,x2+8*x))=−4

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