Math Problem Statement

Calculate the limit

Solution

To find the limit

limxx335x3\lim_{x \to \infty} \frac{x^3 - 3}{5x^3}

we can simplify the expression by dividing both the numerator and the denominator by x3x^3:

limxx335x3=limxx3/x33/x35x3/x3\lim_{x \to \infty} \frac{x^3 - 3}{5x^3} = \lim_{x \to \infty} \frac{x^3/x^3 - 3/x^3}{5x^3/x^3}

This simplifies to:

limx13x35\lim_{x \to \infty} \frac{1 - \frac{3}{x^3}}{5}

As xx approaches infinity, the term 3x3\frac{3}{x^3} approaches 0. So, the expression further simplifies to:

105=15\frac{1 - 0}{5} = \frac{1}{5}

Thus, the limit is:

limxx335x3=15\lim_{x \to \infty} \frac{x^3 - 3}{5x^3} = \frac{1}{5}

Would you like more details or have any further questions?


Related Questions:

  1. How do you find limits at infinity for polynomial functions?
  2. What is L'Hôpital's Rule, and when can it be applied?
  3. How do you handle limits involving indeterminate forms?
  4. Can you explain the difference between limits at infinity and limits as xx approaches a finite number?
  5. What is the significance of horizontal asymptotes in limits at infinity?
  6. How does the degree of the numerator compare to the degree of the denominator in determining the limit at infinity?
  7. What happens if the degrees of the numerator and the denominator are equal?
  8. How do you interpret a limit that equals infinity?

Tip: Simplifying the expression by dividing both numerator and denominator by the highest power of xx is often a good first step when dealing with limits at infinity.

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Math Problem Analysis

Mathematical Concepts

Limits
Infinity
Polynomial Functions

Formulas

Limit definition
Simplification techniques

Theorems

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Suitable Grade Level

Advanced High School or College