Epsilon Delta Definition of Limit, graphical concept, counter examples, important proofs of limits

This is a complete lecture on epsilon delta definition of limit, also called precise definition of limit, by Qais Ali Khan. In this lecture Qais Ali Khan completely explains the concept of epsilon delta definition of limit. Epsilon delta definition of limit is also called precise definition of limit. In this lecture Qais Ali Khan starts the topic with a little revision of intuitive definition of limit. First of all Qais Ali Khan gives some information about epsilon delta definition of limit, after the introduction Qais Ali Khan starts the epsilon delta definition of limit. After explaining the complete epsilon delta definition of limit Qais Ali Khan changes the inequalities in the definition in the form of intervals and neighbourhoods. After changing in the interval form Qais Ali Khan gets a strong point that he uses in the graphical concept of precise definition/epsilon delta definition of limit. Then Qais Ali Khan explains the graphical concept of epsilon delta definition and he shows how to get different deltas for every epsilon if the limit exists. After the explanation of existence of limit Qais Ali khan explains how a limit does not exist according to epsilon delta definition. Qais Ali khan explains that when the limit does not exist, there is an epsilon for which we can not find a delta. Furthermore, Qais Ali Khan explains the method to prove limits of functions. First, the limit of a linear function is proved using epsilon delta definition of limit. This proof has two steps, in the first step a suitable delta is guessed and in the next step it is proved that the delta in our previous step works. After the proof of limit of linear function Qais Ali Khan explains the proof of limit of a quadratic function. Proof of this limit is almost same as of the proof of limit of linear function but there is a difference in the method because when proving the quadratic function's limit we assume a positive number C whose value is used in the proof. Value of C does not matter in the proof of epsilon delta definition because its value is solved out at the some stage during the proof. In this way the whole proof is completed. After completing this proof Qais Ali khan has completed the lecture so he concludes the lecture by revising the topics and concepts he covered during the lecture.

0:00 intro
1:50 use of epsilon delta definition
2:52 Epsilon Delta Definition
8:10 Interval form of epsilon delta condition
15:46 Graphical Concept of Epsilon Delta Definition
22:16 Why should "x" and "a" be distinct
25:14 Counter Example: How does limit not exist
33:48 Question 1: proving limit by epsilon delta definition
43:32 Question 2: proving limit of quadratic function
01:00:56 Conclusion