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Show that $ \displaystyle \lim_{t \to a} r(t) = b $ if and only if for every $ \varepsilon > 0 $ there is a number $ \delta > 0 $ such that if $ 0 < \mid t - a \mid < \delta $ then $ \mid r(t) - b \mid < \varepsilon $

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$\lim _{t \rightarrow a} r(t)=b$

Calculus 3

Chapter 13

Vector Functions

Section 1

Vector Functions and Space Curves

Johns Hopkins University

Missouri State University

Baylor University

Boston College

Lectures

03:04

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x. The input of a function is called the argument and the output is called the value. The set of all permitted inputs is called the domain of the function. Similarly, the set of all permissible outputs is called the codomain. The most common symbols used to represent functions in mathematics are f and g. The set of all possible values of a function is called the image of the function, while the set of all functions from a set "A" to a set "B" is called the set of "B"-valued functions or the function space "B"["A"].

08:32

In mathematics, vector calculus is an important part of differential geometry, together with differential topology and differential geometry. It is also a tool used in many parts of physics. It is a collection of techniques to describe and study the properties of vector fields. It is a broad and deep subject that involves many different mathematical techniques.

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Show that $\lim _{t \right…

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Prove the statement using …

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Use the formal definition …

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First, we prove that if the limit T is able to be then for a every Gibson critics and zero, there is a number that is greater than zero are sad. If there is smaller than humans, it's absolutely of T minus a smaller than data. Then absolute value of our human speed is smarter than you're soon. The first oh any Gibson is greater than zero things limit. Rt is likely to be behalf limit. Axity is equal to be one limit. Why two? It's equal to B two. They made thank you is equal to B three. Here are t is equal to excuse me Why Chief 80 and be is equal to be one Me too three then by the definition of the limiter of function. But now from these three limits for any it soon from is the quarter it's gonna over three, right? By the definition of the limits were half there exists that one. That too got three such that yes, zero is more than she was. A It's more than 3 to 1 XT once B one is smaller than the product proud And if zero smaller than T minus a smaller than two white tea minus B to a small son. Absolutely product. The zero smaller than humanness. A smaller than two or three C g. When the speed three is smaller than excellent, then with lot Dad, it's difficult to you. Minimum value of Yeah. 12 yeah, three. Then, if zero is smaller than T minus, pay more than the other to half. Artie might be absolute value. This is equal to negative. 60 minus B one square us Y T minus B two square, us C T minus B three square So this is it's more in the it's a three hip strewn from square. This is equally true. You too. If some square over three, this is smaller than yes, sir. Then we approve. If, for every Epsilon greeted and zero, there is a data such that if zero is smaller than humanness, a smaller than Delta, then our team in space more than it's more than the limit R T is able to be. Yeah, for any tips greater than zero Vegas this data, and if zero is monitored, she might stay small smaller than data. We have rt oneness being smaller instrument than behalf 60 minus B one is smaller than rt minus b So this is more than six months for this same data and same Epson. Why two? I want to speak Chu. Yeah, smother small 13 minus b three. Yes, Mother. It's more then by the definition of the limits of our function we have limit. XT is equal to be one limit Y t is equal to be true. Limit is equal to three So the limit Archie is equal to limit angsty limit Y t limit Say cheese. This is equal to you. Want to be three? This is equal to

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