Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problem 2.42

Chapter 2, Problem 2.42

Limits and Infinity

Find the limits in Exercises 37–46.

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

Verified step by step guidance

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.

Divide every term in the numerator and the denominator by x³, the highest power of x in the denominator.

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 \( \lim_{{x \to \infty}} \frac{{x + 1}}{{12 + \frac{128}{x^3}}} \).

As x approaches infinity, the term \( \frac{128}{x^3} \) approaches 0 because the denominator grows much faster than the numerator.

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Recommended video:

2:02

Simplifying Trig Expressions Example 1

Related Practice

Textbook Question

Limits and Infinity

Find the limits in Exercises 37–46.

2x² + 3

lim -------------

x→⁻∞ 5x² + 7

Textbook Question

Limits as x → ∞ or x → −∞

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.

lim x→∞ (x − 3) / √(4x² + 25)

Textbook Question

Calculating Limits

Find the limits in Exercises 11–22.

limx→−1/2 4x(3x+4)²

Textbook Question

At what points are the functions in Exercises 13–30 continuous?

y = √(2x + 3)

Textbook Question

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limh→0 sin(sin h) / sin h