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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 21
2장, 문제 21

Finding Limits


In Exercises 9–24, find the limit or explain why it does not exist.




lim x →π sin (x/2 + sin x)

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1
Step 1: Understand the problem. We need to find the limit of the function sin(x/2 + sin(x)) as x approaches π.
Step 2: Substitute x = π into the expression inside the sine function. This gives us π/2 + sin(π). Since sin(π) = 0, the expression simplifies to π/2.
Step 3: Evaluate the sine function at π/2. We know that sin(π/2) = 1.
Step 4: Since the function sin(x/2 + sin(x)) is continuous around x = π, the limit as x approaches π is simply the value of the function at x = π.
Step 5: Conclude that the limit of sin(x/2 + sin(x)) as x approaches π is 1, based on the continuity and substitution.

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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, which is crucial for defining derivatives and integrals. Limits can exist or not exist, and determining their existence often involves evaluating the function at points close to the limit.
추천 영상:
05:50
One-Sided Limits

Continuity

Continuity refers to a property of functions where they do not have any abrupt changes, jumps, or holes at a given point. A function is continuous at a point if the limit as the input approaches that point equals the function's value at that point. Understanding continuity is essential for evaluating limits, as discontinuities can lead to limits that do not exist.
추천 영상:
05:34
Intro to Continuity

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. In the context of limits, these functions can exhibit specific behaviors as their arguments approach certain values, which can affect the limit's existence. Familiarity with the properties and values of trigonometric functions is crucial for evaluating limits involving them.
추천 영상:
6:04
Introduction to Trigonometric Functions
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교과서 질문

[Technology Exercise] Let f(t) = 1/t for t≠0.

         

a. Find the average rate of change of f with respect to t over the intervals (i) from t=2 to t=3, and (ii) from t=2 to t=T.

         

b. Make a table of values of the average rate of change of f with respect to t over the interval [2,T], for some values of T approaching 2, say T = 2.1, 2.01, 2.001, 2.0001, 2.00001, and 2.000001.

         

c. What does your table indicate is the rate of change of f with respect to t at t=2?

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교과서 질문

The accompanying graph shows the total distance s traveled by a bicyclist after t hours.

b. Estimate the bicyclist’s instantaneous speed at the times t=1/2, t=2, and t=3.

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교과서 질문

Finding Limits

In Exercises 9–24, find the limit or explain why it does not exist.



lim x→a (x² ― a²)/(x⁴ ― a⁴)

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교과서 질문

Finding Limits

In Exercises 9–24, find the limit or explain why it does not exist.



lim x →π cos² (x― tan x)

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교과서 질문

The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 80 m to the surface of the moon.

         

a. Estimate the slopes of the secant lines PQ₁, PQ₂, PQ₃, and PQ₄, arranging them in a table like the one in Figure 2.6.


b. About how fast was the object going when it hit the surface?


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교과서 질문

Finding Limits

In Exercises 9–24, find the limit or explain why it does not exist.



lim h →0 ((x + h)² ― x²)/h

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