Chapter 11: Q. 5 (page 859)

Explain why we do not need an 鈥渆psilon鈥揹elta鈥� definition for the limit of a vector-valued function.

Short Answer

Expert verified

$\lim_{t 鈫� c} x (t) = L means for 蔚 > 0, there exists a 未 > 0 such that |x (t) - L| < 蔚 whenever x 鈭� E and 0 < |t - c| < 蔚 . In this case, We say that the limit of the function x (t) as t tends to' c' exists and it is' L' and wee write it as \lim_{t 鈫� c} x (t) = L$

Step by step solution

Step 1. Given information

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

Step 2. Limit of a vector valued function

$Let r (t) = <x (t), y (t), z (t)> be a vector - valued function . The limit of r (t) is t approaches c, denoted by \lim_{t 鈫� c} r (t), is defined by \lim_{t 鈫� c} r (t) = \lim_{t 鈫� c} <x (t), y (t), z (t)> = \lim_{t 鈫� c} x (t) i + \lim_{t 鈫� c} y (t) j + \lim_{t 鈫� c} z (t) k, where \lim_{t 鈫� c} x (t), \lim_{t 鈫� c} y (t) and \lim_{t 鈫� c} z (t) exists .$

Step 3. Result

$We observed that this definition is the sum of the limits of the components of the function . Each component is a function of a single variable . We know that these limits are defined using an " epsilon - delta " definition . Note : \lim_{t 鈫� c} x (t) = L means for 蔚 > 0, there exists a 未 > 0 such that |x (t) - L| < 蔚 whenever x 鈭� E and 0 < |t - c| < 蔚 . In this case, We say that the limit of the function x (t) as t tends to' c' exists and it is' L' and wee write it as \lim_{t 鈫� c} x (t) = L$

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91影视

Explain why we do not need an 鈥渆psilon鈥揹elta鈥� definition for the limit of a vector-valued function.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Limit of a vector valued function

Step 3. Result

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