Evaluate the Limit

Problem

(lim_x→∞)((5*x3+1)/(x−x3))

Solution

1. Identify the highest power of x in the denominator, which is x3

2. Divide both the numerator and the denominator by x3 to simplify the expression for the limit at infinity.

3. Rewrite the expression by distributing x3 to each term.

(lim_x→∞)(((5*x3)/(x3)+1/(x3))/(x/(x3)−(x3)/(x3)))

4. Simplify each term within the fraction.

(lim_x→∞)((5+1/(x3))/(1/(x2)−1))

5. Apply the limit property that (lim_x→∞)(1/(xn))=0 for any n>0

(5+0)/(0−1)

6. Calculate the final numerical value.

5/(−1)=−5

Final Answer

(lim_x→∞)((5*x3+1)/(x−x3))=−5

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