Evaluate the Limit

Problem

(lim_x→−∞)(x+√(,x2+6*x))

Solution

1. Identify the indeterminate form by observing that as x→−∞ the expression x+√(,x2+6*x) results in the form −∞+∞

2. Rationalize the expression by multiplying and dividing by the conjugate, x−√(,x2+6*x)

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

x2−(x2+6*x)=−6*x

4. Rewrite the limit expression with the new denominator.

(lim_x→−∞)((−6*x)/(x−√(,x2+6*x)))

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

√(,x2+6*x)=√(,x2*(1+6/x))=−x√(,1+6/x)

6. Substitute this back into the denominator and factor out x

x−(−x√(,1+6/x))=x*(1+√(,1+6/x))

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

(lim_x→−∞)((−6)/(1+√(,1+6/x)))

8. Apply the limit property (lim_x→−∞)(6/x)=0

(−6)/(1+√(,1+0))=(−6)/2=−3

Final Answer

(lim_x→−∞)(x+√(,x2+6*x))=−3

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