교과서 질문
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limt→0 sin(1 − cos t) / (1 − cos t)
310
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Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
2장, 문제 2.42
Limits and Infinity
Find the limits in Exercises 37–46.
x⁴ + x³
lim -----------------
x→∞ 12x³ + 128
검증된 단계별 안내1
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.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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.
추천 영상:
05:50
One-Sided Limits
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.
추천 영상:
6:04Intro to Rational Functions
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.
추천 영상:
2:02Simplifying Trig Expressions Example 1
관련 실천
교과서 질문
Finding Limits
In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.
lim (4g(x))¹/³ = 2
x →0
302
views
교과서 질문
Limits of quotients
Find the limits in Exercises 23–42.
limx→−1 (√(x² + 8) − 3) / (x + 1)
196
views
교과서 질문
At what points are the functions in Exercises 13–30 continuous?
f(x) = { (x³ − 8)/(x² − 4), x ≠ 2, x ≠ −2
3, x = 2
4, x = −2
209
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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) = x², x = 1
333
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교과서 질문
Limits with trigonometric functions
Find the limits in Exercises 43–50.
limx→−π √(x + 4) cos(x + π)
262
views