Evaluate the Limit

Problem

(lim_x→∞)((5*x^3+1)/(x−x^3))

Solution

1. Identify the highest power of x in the denominator, which is x^3

2. Divide both the numerator and the denominator by x^3 to simplify the expression for the limit at infinity.

3. Rewrite the expression by distributing x^3 to each term.

(lim_x→∞)(((5*x^3)/(x^3)+1/(x^3))/(x/(x^3)−(x^3)/(x^3)))

4. Simplify each term within the fraction.

(lim_x→∞)((5+1/(x^3))/(1/(x^2)−1))

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

(5+0)/(0−1)

6. Calculate the final numerical value.

5/(−1)=−5

Final Answer

(lim_x→∞)((5*x^3+1)/(x−x^3))=−5

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