温州数字教育资源网-温州市教育局-可汗学院：微积分-微分学-0004.极限例题1

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00:00:01

Introduction to Limit Problems

This section introduces the general concept of solving limit problems, emphasizing how limits differ from directly substituting values into a function. It sets up the intuition for evaluating undefined expressions.
00:00:15

Evaluating a Specific Limit

The video begins working through a specific example: finding the limit as x approaches -1 of (2x + 2)/(x + 1). The initial substitution leads to an undefined form 0/0, prompting further analysis.
00:02:11

Simplifying the Expression

This section focuses on simplifying the expression algebraically by factoring out common terms. It explains how the function behaves similarly to a constant function (f(x) = 2), except at x = -1 where it is undefined.
00:03:32

Graphical Representation and Intuition

This part visually demonstrates the behavior of the function using a graph. It highlights that even though the function is undefined at x = -1, the limit still exists and equals 2 based on the surrounding values.
00:05:49

Numerical Method to Verify the Limit

This section discusses alternative ways to verify the limit numerically by plugging in values very close to -1 from both sides. It also suggests using a calculator for practical evaluation.
00:06:27

Another Limit Example: 1/x as x Approaches 0

A new problem is introduced: finding the limit of 1/x as x approaches 0. This section starts exploring this by creating a table of values and hints at the need for graphical interpretation due to diverging behavior.

本视频通过具体例题讲解了如何求解函数的极限，包括图像分析和代数化简的方法，并介绍了在遇到未定型时如何通过简化表达式或代入近似值来确定极限值。

课程名称：可汗学院：微积分-微分学

责任教师：管理员单位机构：温州市教育局

课程简介：

姓名	所在单位	学科/专业
管理员	温州市教育局

课程章节
课程资料

自动连播

0001.牛顿，莱布尼茨和尤塞恩·博尔特

0002.极限介绍1

0003.极限介绍2

0004.极限例题1

0005.极限例题2

0006.极限例题3

0007.极限例题4

0008.夹逼定理

0009.证明sinx除以x的极限

0010.更多极限例题

0011.极限的ε-δ定义1

0012.极限的ε-δ定义2

0013.导数(新版本)1

0014.导数(新版本)2

0015.导数(新版本)2.5

0016.直观导数模块

0017.导数1

0018.导数2

0019.导数3

0020.链式法则

0021.链式法则例题

0022.更多链式法则的例题

0023.乘积法则

0024.商法则

0025.导数9

0026.证明：ddx(x^n)

0027.证明：ddx(sqrt(x))

0028.证明：ddx(ln x) = 1x

0029.证明：ddx(e^x) = e^x

0030.证明ln(x)和e^x的导数

0031.31.extreme derivative word problem (advanced)

0032.32.implicit differentiation

0033.33.implicit differentiation (part 2)

0034.34.more implicit differentiation

0035.35.more chain rule and implicit differentiation intuition

0036.36.trig implicit differentiation example

0037.37.calculus derivative of x^(x^x)

0038.38.introduction to l'hopital's rule

0039.39.l'hopital's rule example 1

0040.40.l'hopital's rule example 2

0041.41.l'hopital's rule example 3

0042.42.maxima minima slope intuition

0043.43.inflection points and concavity intuition

0044.44.monotonicity theorem

0045.45.calculus maximum and minimum values on an interval

0046.46.calculus graphing using derivatives

0047.47.calculus graphing with derivatives example

0048.48.graphing with calculus

0049.49.optimization with calculus 1

0050.50.optimization with calculus 2

0051.51.optimization with calculus 3

0052.52.optimization example 4

0053.53.introduction to rate-of-change problems

0054.54.equation of a tangent line

0055.55.rates-of-change (part 2)

0056.56.ladder rate-of-change problem

0057.57mean value theorem

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