Understanding Precise Definitions of Limits in Analysis
Delve into the precise definitions of limits in mathematical analysis, exploring left-hand limits, right-hand limits, limit of a function, and more. Gain insights into how limits are defined and understood in the context of mathematical functions.
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MAT 3749 Introduction to Analysis Section 2.1 Part 1 Precise Definition of Limits http://myhome.spu.edu/lauw
References Section 2.1
Preview The elementary definitions of limits are vague. Precise (? ?) definition for limits of functions. The definition is highly polished and stated sophisticatedly. It takes time to get used to.
Recall: Left-Hand Limit We write = lim x ( ) f x L a and say the left-hand limit of ?(?), as ? approaches ?, equals ? if we can make the values of ?(?) arbitrarily close to ? (as close as we like) by taking ? to be sufficiently close to ? and ? less than a.
Recall: Right-Hand Limit We write = lim x ( ) f x L + a and say the right-hand limit of ?(?), as ? approaches ?, equals ? if we can make the values of ?(?) arbitrarily close to ? (as close as we like) by taking ? to be sufficiently close to ? and ? greater than a.
Recall: Limit of a Function = lim ( ) f x L x a if and only if = = lim ( ) f x L lim ( ) f x L and + x a x a Independent of ?(?)
Recall 1 x limsin x DNE 0
Deleted Neighborhood For all practical purposes, we are going to use the following definition:
Precise Definition L L L L L a a a a a
Precise Definition L L L L L a a a a a
Precise Definition if we can make the values of ?(?) arbitrarily close to ? (as close as we like) by taking ? to be sufficiently close toa
Precise Definition ( ) f x Given we want are within a distance of and L if we can make the values of ?(?) arbitrarily close to ? (as close as we like) by taking ? to be sufficiently close toa
Precise Definition ( ) f x Given we want are within a distance of and L if we can make the values of ?(?) arbitrarily close to ? (as close as we like) by taking ? to be sufficiently close toa we can find a , such that and are within a distance of x a
Precise Definition The values of ? depend on the values of ?. ? is a function of ?. Definition is independent of the function value at a
Example 1 Use the ? ? definition to prove that ( 1 x ) + = lim 2 3 5 x
Analysis Use the ? ? definition to prove that ( 1 x ) + = lim 2 3 5 x
Proof Use the ? ? definition to prove that ( 1 x ) + = lim 2 3 5 x
Guessing and Proofing Note that in the solution of Example 1 there were two stages guessing and proving. We made a preliminary analysis that enabled us to guess a value for ?. But then in the second stage we had to go back and prove in a careful, logical fashion that we had made a correct guess. This procedure is typical of much of mathematics. Sometimes it is necessary to first make an intelligent guess about the answer to a problem and then prove that the guess is correct.
Remarks We are in a process of reinventing calculus. Many results that you have learned in calculus courses are not yet available! It is like
Remarks Most limits are not as straight forward as Example 1.
Example 2 Use the ? ?definition to prove that lim x = 2 9 x 3
Analysis Use the ? ?definition to prove that lim x = 2 9 x 3
Analysis Use the ? ?definition to prove that lim x = 2 9 x 3
Proof Use the ? ?definition to prove that lim x = 2 9 x 3
Questions 1 What is so magical about 1 in ? Can this limit be proved without using the minimum technique? min ,7
Example 3 1 x Prove that limsin x DNE 0
Analysis 1 x Prove that limsin x DNE 0
Proof (Classwork) Prove that 1 x limsin x DNE 0
