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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Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
2장, 문제 13
Finding Limits
In Exercises 9–24, find the limit or explain why it does not exist.
lim h →0 ((x + h)² ― x²)/h
검증된 단계별 안내1
Recognize that the given expression is a difference quotient, which is often used to find the derivative of a function at a point. Here, the function is f(x) = x².
Substitute the expression (x + h)² for f(x + h) in the difference quotient: ((x + h)² - x²)/h.
Expand the expression (x + h)² using the binomial theorem: (x + h)² = x² + 2xh + h².
Substitute the expanded form back into the difference quotient: ((x² + 2xh + h²) - x²)/h.
Simplify the expression by canceling out x² and then dividing each term by h: (2xh + h²)/h = 2x + h. Now, take the limit as h approaches 0, which results in the derivative of x² at x.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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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. In this case, we are interested in the limit as h approaches 0, which is essential for evaluating the expression given.
추천 영상:
05:50
One-Sided Limits
Difference Quotient
The difference quotient is a formula that represents the average rate of change of a function over an interval. It is expressed as (f(x + h) - f(x)) / h, and is used to derive the derivative of a function. In the given limit problem, the expression ((x + h)² - x²)/h is a difference quotient that simplifies to find the derivative of the function f(x) = x².
추천 영상:
06:43The Quotient Rule
Derivative
The derivative of a function at a point measures the instantaneous rate of change of the function at that point. It is defined as the limit of the difference quotient as the interval approaches zero. In this exercise, finding the limit will ultimately yield the derivative of the function f(x) = x², which is a key application of limits in calculus.
추천 영상:
05:44Derivatives
관련 실천
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)
210
views
교과서 질문
Limits and Continuity
In Exercises 5 and 6, find the value that lim (x→0) g(x) must have if the given limit statements hold.
lim (x lim g(x)) = 2
x→-4 x→0
188
views
교과서 질문
Finding Deltas Graphically
In Exercises 7–14, use the graphs to find a δ>0 such that |f(x)−L| <ε whenever 0< |x−c| <δ.
379
views
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?
290
views
교과서 질문
Finding Limits
In Exercises 9–24, find the limit or explain why it does not exist.
lim x →π sin (x/2 + sin x)
285
views