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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 2.4.27
2장, 문제 2.4.27

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limx→0 (tan 2x) / x

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1
Recognize that the limit involves the function tan(2x), which can be rewritten using the identity tan(2x) = sin(2x) / cos(2x). This gives us the expression lim(x→0) (sin(2x) / (x * cos(2x))).
To simplify the expression, focus on the term sin(2x) / x. Notice that this can be rewritten as (sin(2x) / 2x) * (2x / x).
Apply the limit property lim(θ→0) (sin(θ) / θ) = 1 to the term (sin(2x) / 2x). This implies that as x approaches 0, (sin(2x) / 2x) approaches 1.
Now, consider the term (2x / x), which simplifies to 2. Therefore, the expression (sin(2x) / x) simplifies to 2 * (sin(2x) / 2x).
Finally, evaluate the limit of the entire expression: lim(x→0) (2 * (sin(2x) / 2x) / cos(2x)). Since (sin(2x) / 2x) approaches 1 and cos(2x) approaches cos(0) = 1 as x approaches 0, the limit simplifies to 2 * 1 / 1.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Limit of a Function

The limit of a function describes the behavior of the function as the input approaches a certain value. In calculus, understanding limits is crucial for analyzing how functions behave near specific points, which is foundational for defining derivatives and integrals. The limit limθ→0 sin θ / θ = 1 is a standard result used to evaluate limits involving trigonometric functions.
추천 영상:
06:11
Limits of Rational Functions: Denominator = 0

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. They are essential tools for simplifying expressions and solving problems in calculus. For example, the identity tan x = sin x / cos x can be used to rewrite and simplify the expression tan 2x in terms of sine and cosine, aiding in the evaluation of limits.
추천 영상:
7:17
Verifying Trig Equations as Identities

L'Hôpital's Rule

L'Hôpital's Rule is a method for finding limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) as x approaches a point results in an indeterminate form, the limit can be found by differentiating the numerator and denominator separately. This rule is particularly useful for evaluating limits involving complex expressions, such as limx→0 (tan 2x) / x.
추천 영상:
가이드 코스
5:50
Power Rules
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교과서 질문

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→⁻∞ (³√x − ⁵√x) / (³√x + ⁵√x)

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교과서 질문

At what points are the functions in Exercises 13–30 continuous?


y = cos (x) / x

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교과서 질문

Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?


lim x → 0 tan (π/4 cos (sin x¹/³))

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교과서 질문

Finding Limits

In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.


5 ―x²

lim ------------- = 0

x → ―2 (√g(x))


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교과서 질문

Limits of quotients


Find the limits in Exercises 23–42.


limx→−3 (2 − √(x² − 5)) / (x + 3)

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교과서 질문

[Technology Exercise] 22. Make a table of values for the function at the points x=1.2, x=11/10, x=101/100, x=1001/1000, x=10001/10000, and x = 1.

         

a. Find the average rate of change of F(x) over the intervals [1,x] for each x≠1 in your table.

         

b. Extending the table if necessary, try to determine the rate of change of F(x) at x = 1.

335
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