Q. 5

Question

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

Step-by-Step Solution

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Answer

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

1Step 1. Given information

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

2Step 2. Limit of a vector valued function

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

3Step 3. Result

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

Q. 4TB

Q. 6

Other exercises in this chapter

Q. 4

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

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Q. 4TB

Parametric equations for a circle: Find parametric equations whose graph is the circle with radius ρ centered at the point (a, b) in the xy-plane such that

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Q. 6

Let y = f(x). State the definition for the continuity of the function f at a point c in the domain of f .

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Q. 6

Let y = f (x). State the definition for the continuity of the function f at a point c in the domain of f .

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