Skip to main content
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 2d
2장, 문제 2d

Finding Limits Graphically


Which of the following statements about the function y = f(x) graphed here are true, and which are false?


img


d. limx→1− f(x) = 2

검증된 단계별 안내
1
To determine the limit \( \lim_{x \to 1^-} f(x) = 2 \), we need to analyze the behavior of the function \( f(x) \) as \( x \) approaches 1 from the left side (i.e., values of \( x \) that are slightly less than 1).
Examine the graph of the function \( y = f(x) \) and observe the values of \( f(x) \) as \( x \) gets closer to 1 from the left. Look for the y-values that the function approaches.
Identify the y-value that \( f(x) \) is approaching as \( x \) approaches 1 from the left. This is the left-hand limit, denoted as \( \lim_{x \to 1^-} f(x) \).
Check if the y-value that \( f(x) \) approaches from the left is equal to 2. If the graph shows that the function approaches a y-value of 2 as \( x \) approaches 1 from the left, then the statement is true.
If the y-value that \( f(x) \) approaches from the left is not equal to 2, then the statement is false. Conclude based on the observed behavior of the graph.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
3m
도움이 되었나요?

주요 개념

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

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, even if they are not defined at those points. For example, the limit of f(x) as x approaches 1 from the left (denoted as lim x→1− f(x)) examines the values f(x) takes as x gets closer to 1 from values less than 1.
추천 영상:
05:50
One-Sided Limits

One-Sided Limits

One-sided limits refer to the limits of a function as the input approaches a specific value from one side only. The left-hand limit (lim x→c− f(x)) considers values approaching c from the left, while the right-hand limit (lim x→c+ f(x)) considers values approaching c from the right. Understanding one-sided limits is crucial for analyzing functions that may have different behaviors on either side of a point.
추천 영상:
05:50
One-Sided Limits

Graphical Interpretation of Limits

Graphical interpretation of limits involves analyzing the graph of a function to determine the value that the function approaches as the input approaches a certain point. By observing the behavior of the graph near that point, one can visually assess whether the limit exists and what its value is. This method is particularly useful for identifying discontinuities or jumps in the function's values.
추천 영상:
6:47
Finding Limits Numerically and Graphically
관련 실천
교과서 질문

Horizontal and Vertical Asymptotes


Use limits to determine the equations for all horizontal asymptotes.

_________

/ x² + 9

d. y = / -------------

√ 9x² + 1

225
views
교과서 질문

Limits and Continuity

On what intervals are the following functions continuous?


d. k(x) = sin x / x

229
views
교과서 질문

Limits and Continuity


On what intervals are the following functions continuous?


c. h(x) = cos x / x―π

221
views
교과서 질문

Finding Limits Graphically


Which of the following statements about the function y = f(x) graphed here are true, and which are false?



e. limx→1+ f(x) = 1

338
views
교과서 질문

Finding Limits Graphically


Graph the functions in Exercises 9 and 10. Then answer these questions.


f(x) = {x,−1 ≤ x < 0, or 0 < x ≤ 1

1, x = 0

0, x < −1 or x > 1


d. At what points does the right-hand limit exist but not the left-hand limit?

311
views
교과서 질문

Limits and Continuity

On what intervals are the following functions continuous?


d. k(x) = x⁻¹/⁶

239
views