Evaluate the Limit

Problem

(lim_x→−∞)(x8+x9)

Solution

1. Identify the highest power of x in the polynomial to determine the end behavior as x approaches −∞

2. Factor out the highest power, x9 from the expression to analyze the limit of each component.

(lim_x→−∞)(x9)*(x(−1)+1)

3. Simplify the expression inside the parentheses by rewriting x(−1) as a fraction.

(lim_x→−∞)(x9)*(1/x+1)

4. Evaluate the limit of each part as x→−∞ Note that 1/x approaches 0

(lim_x→−∞)(1/x+1)=0+1=1

5. Determine the limit of the leading term x9 Since the exponent is odd, as x approaches negative infinity, x9 approaches negative infinity.

(lim_x→−∞)(x9)=−∞

6. Multiply the results of the limits. The product of a term approaching −∞ and a term approaching 1 is −∞

(−∞)⋅(1)=−∞

Final Answer

(lim_x→−∞)(x8+x9)=−∞

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