Let r(t) = x(t),y(t),z(t). What is the definition of limt→cr(t)?

limt→cr(t) = limt→cx(t)i + limt→cy(t)j + limt→cz(t)k, which is a vector.

Q. 4 - Calculus | StudyQuestionHub

Q. 4

Question

$Let r (t) = <x (t), y (t), z (t)> . What is the definition of \lim_{t \to c} r (t) ?$

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Answer

$\lim_{t \to c} r (t) = \lim_{t \to c} x (t) i + \lim_{t \to c} y (t) j + \lim_{t \to c} z (t) k, which is a vector .$

1Step 1. Given information is

$r (t) = <x (t), y (t), z (t)>$

2Step 2. Definition of lim t → c r ( t )

$The limit of r (t) as t approaches c, denoted by \lim_{t \to c} r (t), is defined by \lim_{t \to c} r (t) = \lim_{t \to c} <x (t), y (t), z (t)> = \lim_{t \to c} x (t) i + \lim_{t \to c} y (t) j + \lim_{t \to c} z (t) k, where \lim_{t \to c} x (t), \lim_{t \to c} y (t) and \lim_{t \to c} z (t) exists . Thus \lim_{t \to c} r (t) = \lim_{t \to c} x (t) i + \lim_{t \to c} y (t) j + \lim_{t \to c} z (t) k, which is a vector .$

Q. 3TB

Q. 4TB

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Q. 3TB

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Q. 4TB

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Q. 5

Explain why we do not need an “epsilon–delta” definition for the limit of a vector-valued function.

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