Ch. 2 - Limits and Continuity

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

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problem 2.38

Chapter 2, Problem 2.38

Limits and Infinity

Find the limits in Exercises 37–46.

2x² + 3
lim -------------
x→⁻∞ 5x² + 7

Verified step by step guidance

Identify the highest degree terms in the numerator and the denominator. In this case, both the numerator and the denominator have the highest degree term of x².

Rewrite the limit expression by factoring out the highest degree term from both the numerator and the denominator. This gives us: lim (x→⁻∞) [(2x²/x²) + (3/x²)] / [(5x²/x²) + (7/x²)].

Simplify the expression by canceling out the x² terms in the numerator and the denominator. This results in: lim (x→⁻∞) [2 + (3/x²)] / [5 + (7/x²)].

Evaluate the limit as x approaches negative infinity. As x becomes very large in magnitude, the terms (3/x²) and (7/x²) approach zero.

Conclude that the limit is the ratio of the leading coefficients of the highest degree terms, which is 2/5.

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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. They help in understanding the behavior of functions at specific points, including points of discontinuity or infinity. In this context, we are interested in the limit of a rational function as x approaches negative infinity.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They can exhibit different behaviors as the variable approaches certain values, including infinity. Analyzing the leading terms of the numerator and denominator is crucial for determining the limit of a rational function at infinity.

Behavior at Infinity

The behavior of functions as they approach infinity is essential for evaluating limits. For rational functions, this often involves comparing the degrees of the polynomials in the numerator and denominator. If the degrees are the same, the limit is the ratio of the leading coefficients; if the degree of the numerator is less, the limit approaches zero.

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Related Practice

Textbook Question

Limits of Average Rates of Change

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.

f(x) = 1/x, x = -2

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Textbook Question

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx → √3 1/x² = 1/3

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Textbook Question

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limh→0− h / sin 3h

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Textbook Question

In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)

lim x → ±∞ f(x) = 0, lim x → 2⁻ f(x) = ∞, and lim x → 2⁺ f(x) = ∞

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Textbook Question

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→0 x² sin (1/x) = 0

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Textbook Question

Finding Limits of Differences When x → ±∞