Ch. 2 - Limits and Continuity

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

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problem 2.6.23

Chapter 2, Problem 2.6.23

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 → ∞ √((8x² − 3) / (2x² + x))

Verified step by step guidance

Identify the highest power of x in the denominator. In this case, the highest power is x².

Divide both the numerator and the denominator by x². This simplifies the expression to √((8 - 3/x²) / (2 + 1/x)).

As x approaches ∞, the terms 3/x² and 1/x approach 0 because they contain x in the denominator, which becomes very large.

Substitute these limits into the simplified expression: √((8 - 0) / (2 + 0)) = √(8/2).

Simplify the expression inside the square root: √(8/2) = √4.

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Key Concepts

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

Limits at Infinity

Limits at infinity involve finding the behavior of a function as the variable approaches positive or negative infinity. This often requires simplifying the function to identify dominant terms, which dictate the function's end behavior. Understanding limits at infinity is crucial for analyzing asymptotic behavior and determining horizontal asymptotes of rational functions.

Rational Functions

A rational function is a ratio of two polynomials. To find limits involving rational functions, especially as x approaches infinity, it's essential to compare the degrees of the numerator and denominator. The highest degree terms often dominate the behavior of the function, simplifying the process of finding limits at infinity.

Simplifying Expressions

Simplifying expressions, particularly those involving square roots and polynomials, is key to solving limit problems. By dividing the numerator and denominator by the highest power of x in the denominator, we can reduce the expression to a simpler form. This technique helps in identifying the leading terms and determining the limit as x approaches infinity.

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Related Practice

Textbook Question

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0⁺ 1 / 3x

Textbook Question

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→1 1/x = 1

Textbook Question

Use formal definitions to prove the limit statements in Exercises 93–96.

lim x → 3 (−2 / (x − 3)²) = −∞

Textbook Question

The sign-preserving property of continuous functions Let f be defined on an interval (a, b) and suppose that f(c) ≠ 0 at some c where f is continuous. Show that there is an interval (c − δ, c + δ) about c where f has the same sign as f(c).

Textbook Question

Suppose that f(x) and g(x) are polynomials in x. Can the graph of f(x)/g(x) have an asymptote if g(x) is never zero? Give reasons for your answer.

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Textbook Question

Never-zero continuous functions Is it true that a continuous function that is never zero on an interval never changes sign on that interval? Give reasons for your answer.