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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 2 - Limits and Continuity문제 2.38
2장, 문제 2.38

Limits and Infinity


Find the limits in Exercises 37–46.


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

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1
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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주요 개념

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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.
추천 영상:
05:50
One-Sided Limits

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.
추천 영상:
6:04
Intro to Rational Functions

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.
추천 영상:
03:07
Cases Where Limits Do Not Exist
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교과서 질문

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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교과서 질문

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


limx → √3 1/x² = 1/3

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교과서 질문

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limh→0− h / sin 3h

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교과서 질문

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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교과서 질문

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


lim x→0 x² sin (1/x) = 0


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교과서 질문

Finding Limits of Differences When x → ±∞


Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)


lim x → ∞ (√(9x² − x) − 3x)

270
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