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WZ
Numerade Educator

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Problem 53 Hard Difficulty

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) $

Answer

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

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Video Transcript

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