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Suppose $ u $ and $ v $ are vector functions that possess limits as $ t \to a $ and let $ c $ be a constant. Prove the following properties of limits.(a) $ \displaystyle \lim_{t \to a} [u(t) + v(t)] = \displaystyle \lim_{t \to a} u(t) + \displaystyle \lim_{t \to a} v(t) $(b) $ \displaystyle \lim_{t \to a} cu(t) = c \displaystyle \lim_{t \to a} u(t) $(c) $ \displaystyle \lim_{t \to a} [u(t) \cdot v(t)] = \displaystyle \lim_{t \to a} u(t) \cdot \displaystyle \lim_{t \to a} v(t) $(d) $ \displaystyle \lim_{t \to a} [u(t) \times v(t)] = \displaystyle \lim_{t \to a} u(t) \times \displaystyle \lim_{t \to a} v(t) $

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a. See work for answerb. $\lim _{x \rightarrow a}[c \cdot f(x)]=c \lim _{x \rightarrow a} f(x)$c. see work for answerd. see work for answer

Calculus 3

Chapter 13

Vector Functions

Section 1

Vector Functions and Space Curves

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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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The problem is suppose you and we work the functions at precise limits, as goes to a and let c be a constant prove the following properties of limit limit equal to a? U t, plus t is equal to limit. U t plus limit of 80 point the first lot: u t is equal to x, 1 t x y 1, t th, 1 t and v. T is equal to x, 2 t y 2 t 2 g, then for 80 ut prosperity. This is equal to x, 1 plus x, 2, t y 1 t plus y 2 to 31 t plus t 2 t. Then the lintot goes to a. U t, plus 80 point is equal to limit equals to a x t x, 1 t plus x, 2. T limited to a y 1 t, plus y 2 to limit as to 81 p s, t 2 t this is equal to limit goes to a x. 1 has limits to a x 2 t limitequals to a y 1 t plus limit gas to a y 2 timid, equals to a t o t s limits 22. This is equal to limit x on the mole welimit 313 plus we met x 2 t limit y 2 limit equal to limit equals to 8 x 1 t y over z, 1 t plus limit x 2 t y 2 t z, 2 tis equal to limit Cross to a: u t, plus limit cross to so use the same method, we can solve the next 3 equations.

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