Limits and Infinity


Find the limits in Exercises 37–46.


x⁴ + x³
lim -----------------
x→∞ 12x³ + 128

Verified step by step guidance

1. Identify the highest power of x in both the numerator and the denominator. In this case, the highest power in the numerator is x⁴ and in the denominator is x³.
2. Divide every term in the numerator and the denominator by x³, the highest power of x in the denominator.
3. Rewrite the expression: \( \lim_{{x \to \infty}} \frac{{x^4/x^3 + x^3/x^3}}{{12x^3/x^3 + 128/x^3}} \). This simplifies to \( \lim_{{x \to \infty}} \frac{{x + 1}}{{12 + \frac{128}{x^3}}} \).
4. As x approaches infinity, the term \( \frac{128}{x^3} \) approaches 0 because the denominator grows much faster than the numerator.
5. Evaluate the limit: \( \lim_{{x \to \infty}} \frac{{x + 1}}{{12}} \). As x approaches infinity, the dominant term is x, so the limit simplifies to \( \frac{x}{12} \), which approaches infinity.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, we are interested in the behavior of the function as x approaches infinity, which helps us understand the end behavior of rational functions.

Rational Functions

A rational function is a ratio of two polynomials. In the given limit problem, the numerator is a polynomial of degree 4, and the denominator is a polynomial of degree 3. The degrees of the polynomials play a crucial role in determining the limit as x approaches infinity.

Dominant Terms

In limits involving polynomials, the dominant term is the term with the highest degree, as it has the most significant impact on the function's value as x approaches infinity. For the given expression, the dominant term in the numerator is x⁴, and in the denominator, it is 12x³, which will dictate the limit's value.