Skip to main content
Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 2 - Limits문제 2.7.17
2장, 문제 2.7.17

Suppose |f(x) − 5|<0.1 whenever 0<x<5. Find all values of δ>0 such that |f(x) − 5|<0.1 whenever 0<|x−2|<δ.

검증된 단계별 안내
1
Step 1: Understand the problem statement. We are given that |f(x) - 5| < 0.1 for 0 < x < 5. We need to find a δ > 0 such that |f(x) - 5| < 0.1 whenever 0 < |x - 2| < δ.
Step 2: Recognize that this is a problem about continuity and limits. Specifically, it is related to the definition of a limit at a point, where we want to ensure that f(x) is close to 5 when x is close to 2.
Step 3: Consider the interval 0 < x < 5. Since we want 0 < |x - 2| < δ, we are focusing on values of x that are close to 2 but still within the interval (0, 5).
Step 4: Choose δ such that the interval (2 - δ, 2 + δ) is contained within (0, 5). This ensures that for any x in (2 - δ, 2 + δ), the condition 0 < x < 5 is satisfied, and thus |f(x) - 5| < 0.1 holds.
Step 5: Determine the largest possible δ by considering the boundaries of the interval (0, 5) and the point x = 2. The largest δ will be the minimum of the distances from 2 to the endpoints of the interval (0, 5).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m
도움이 되었나요?

주요 개념

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

Epsilon-Delta Definition of Limit

The epsilon-delta definition of a limit formalizes the concept of limits in calculus. It states that for every ε (epsilon) greater than 0, there exists a δ (delta) such that if the distance between x and a point a is less than δ, then the distance between f(x) and L (the limit) is less than ε. This definition is crucial for understanding continuity and limits in calculus.
추천 영상:
가이드 코스
05:43
Definition of the Definite Integral

Absolute Value Inequalities

Absolute value inequalities express the distance between two values. The inequality |f(x) - 5| < 0.1 indicates that the function f(x) is within 0.1 units of the value 5. Understanding how to manipulate and interpret these inequalities is essential for solving problems related to limits and continuity.
추천 영상:
가이드 코스
06:37
Average Value of a Function

Neighborhoods in Calculus

In calculus, a neighborhood around a point a is defined as the set of all points within a certain distance from a. Specifically, the notation |x - 2| < δ describes a neighborhood around the point x = 2. This concept is important for establishing conditions under which a function behaves predictably, particularly in the context of limits and continuity.
추천 영상:
가이드 코스
06:11
Fundamental Theorem of Calculus Part 1
관련 실천
교과서 질문

Evaluate each limit. 


limx→πcos2(x)+3cos(x)+2cos(x)+1{\(\displaystyle\]\lim\)_{x\(\to\[\pi\)}\(\frac{\cos^2\left(x\right)+3\cos\left(x\right)+2}{\cos^{}\]\left\)(x\(\right\))+1}}

316
views
교과서 질문

Suppose limx→a f(x)=L{\(\displaystyle\]\lim\)_{x\(\to\) a}}\(\text{ }\)f\(\left\)(x\(\right\))=L. Prove that limx→a (c(f(x))=cL{\(\displaystyle\]\lim\)_{x\(\to\) a}}\(\text{ (}\)c\(\left\)(f\(\left\)(x\(\right\))\(\right\))=cL, where cc is a constant.

368
views
교과서 질문

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.


lim p→2 3p / √4p + 1 − 1

283
views
교과서 질문

Determine the following limits.


limθ→02+sinθ1−cos2θ{\(\displaystyle\[\lim\)_{\(\theta\]\to\)0}}\(\frac{2+\sin\theta}{1-\cos^2\theta}\)

323
views
교과서 질문

Determine the interval(s) on which the following functions are continuous. 

p(x)=4x^5−3x^2+1

350
views
교과서 질문

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right? 


f(z)=(z−1)^3/4

257
views