Ch. 2 - Limits and Continuity

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

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problem 2.6.25

Chapter 2, Problem 2.6.25

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 → ⁻∞ ((1 − x³) / (x² + 7x))⁵

Verified step by step guidance

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

Divide both the numerator and the denominator by x², the highest power of x in the denominator.

Rewrite the expression: ((1/x² - x³/x²) / (1 + 7/x))⁵.

Simplify the expression: ((1/x² - x) / (1 + 7/x))⁵.

Evaluate the limit as x approaches -∞. As x → -∞, 1/x² → 0, 7/x → 0, and -x → ∞. Therefore, the expression simplifies to ((0 - ∞) / (1 + 0))⁵, which further simplifies to (-∞)⁵.

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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 concept helps determine the end behavior of functions, particularly rational functions, by analyzing the dominant terms that influence the function's growth or decay.

Rational Functions

Rational functions are quotients of two polynomials. Understanding their limits involves identifying the highest power of x in the denominator and numerator, which dictates the function's behavior as x approaches infinity. Simplifying by dividing by the highest power of x helps reveal the dominant terms affecting the limit.

Dominant Terms

Dominant terms in a polynomial are those with the highest degree, which significantly influence the function's behavior as x approaches infinity. By focusing on these terms, we can simplify the limit calculation, as lower-degree terms become negligible, allowing us to determine the limit more efficiently.

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Textbook Question

Define h(2) in a way that extends h(t) = (t² + 3t − 10)/(t − 2) to be continuous at t = 2.

Textbook Question

Using Limit Rules

Suppose lim x→0 f(x) = 1 and lim x→0 g(x) = −5. Name the rules in Theorem 1 that are used to accomplish steps (a), (b), and (c) of the following calculation.

limx→0 (2f(x) − g(x)) / (f(x) + 7)² = limx→0 (2f(x) − g(x)) / limx→0 (f(x) + 7)² (a)

(We assume the denominator is nonzero.)

(lim x→0 2f(x) − lim x→0 g(x)) / (lim x→0 (f(x) + 7))² (b)

= (2 lim x→0 f(x) − lim x→0 g(x)) / (lim x→0 f(x) + lim x→0 7)² (c)

= ((2)(1) − (−5)) / (1 + 7)² = 7/64

Textbook Question

For what values of a and b is

g(x) = { ax + 2b, x ≤ 0

x² + 3a – b, 0 < x ≤ 2

3x – 5, x > 2

continuous at every x?

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Textbook Question

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→9 √(x − 5) = 2

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).