Limits of Vector Valued Functions

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Limits of Vector-Valued Functions

From “Limits and Continuity of a Vector-Valued Function” in section 2.1 of the textbook.

Questions?

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Key Ideas

The limit of a vector-valued function is the vector of limits of the components.

Definition

Can you put the formal definition of the limit of a vector-valued function into words? Or into a picture?

As t gets closer to a, the distance between L and the points r(t) gets closer to 0:

Examples

Find lim_{t → 2} ⟨ t, (t² - 2t) / (t-2), sin(πt)/(2-t) ⟩

Take the limits of each component. You can find the limit of the x component by plugging t = 2 into the expression, but for the other components you get 0/0 when you do that. Since you can use L’Hôpital’s rule in that case, we did, and got expressions we could plug t = 2 into:

All the other things you know about limits of scalar functions (e.g., one-sided limits, ways of recognizing that a limit doesn’t exist) also apply to vector-valued functions.

Continuity

Also from “Limits and Continuity of a Vector-Valued Function” in section 2.1 of the textbook.

Questions?

(No.)

Key Ideas

The definition of continuity at a point is the same as for scalar functions (i.e., the limit must exist, the function must have a value, and those 2 things must be equal)

Examples

Is s(t) = ⟨ 2/(1-t), (t² - 2t) / (t-2), |t-1|/(t²-1) ⟩ continuous at t = 1? t = 2?

Find an example of a value for t at which this function is continuous.

Next

Derivatives of vector-valued functions.

Please read “Derivatives of Vector-Valued Functions” and “Tangent Vectors and Unit Tangent Vectors” in section 2.2 of the textbook for Friday.

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