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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 2.2.47
2장, 문제 2.2.47

Limits with trigonometric functions


Find the limits in Exercises 43–50.


lim x→0 tan x

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1
Recognize that the limit involves the trigonometric function tan(x) as x approaches 0.
Recall the identity for tangent: tan(x) = sin(x) / cos(x). This will help in simplifying the expression.
Consider the limit: lim x→0 tan(x) = lim x→0 (sin(x) / cos(x)).
Use the fact that as x approaches 0, sin(x) approaches 0 and cos(x) approaches 1. This simplifies the expression.
Apply the limit: lim x→0 (sin(x) / cos(x)) = (lim x→0 sin(x)) / (lim x→0 cos(x)) = 0 / 1.

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주요 개념

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

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points of discontinuity or where the function is not explicitly defined. Evaluating limits is essential for defining derivatives and integrals.
추천 영상:
05:50
One-Sided Limits

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. They are periodic functions that play a crucial role in calculus, especially when dealing with limits, derivatives, and integrals involving angles. Understanding their properties and behaviors is key to solving limit problems involving these functions.
추천 영상:
6:04
Introduction to Trigonometric Functions

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful when dealing with limits involving trigonometric functions.
추천 영상:
가이드 코스
5:50
Power Rules
관련 실천
교과서 질문

In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?

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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 → −∞ (2x + √(4x² + 3x − 2))

289
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교과서 질문

Using the Sandwich Theorem


If 2−x² ≤ g(x) ≤ 2cosx for all x, find limx→0 g(x).

364
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교과서 질문

Slope of a Curve at a Point


In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.


y=√7−x, P(−2,3)

325
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교과서 질문

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limθ→0 sin θ / sin 2θ

308
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교과서 질문

Finding Deltas Algebraically


Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.


f(x) = 1/x, L = 1/4, c = 4, ε = 0.05

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