Engineering Calculus Notes 176

Engineering Calculus Notes 176 - 164 CHAPTER 2. CURVES −...

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Unformatted text preview: 164 CHAPTER 2. CURVES − → → − Definition 2.3.10. f : R → R3 converges to L ∈ R3 as t → t0 if t0 is an → − accumulation point of the domain of f (t) and for every sequence {ti } in → − the domain of f which converges to, but is distinct from, t0 , the sequence → − → − of points pi = f (ti ) converges to L . We write → − → − f (t) → L as t → t0 or → − → − L = lim f (t) t→t0 when this holds. → − Again, convergence of f relates immediately to convergence of its components: → − → − Remark 2.3.11. f : R → R3 converges to L as t → t0 precisely when the → − → − components of f converge to the components of L as t → t0 . → − If any of the component functions diverges as t → t0 , then so does f (t). The following algebraic properties of limits are easy to check (Exercise 8): → −→ Proposition 2.3.12. Suppose f , − : R → R3 satisfy g → − → − L f = lim f (t) t→t0 → − → g L g = lim − (t) t→t0 and r: R → R satisfies Lr = lim r (t). t→t0 Then 1. lim t→t0 → − → − → − → f (t) ± − (t) = L f ± L g g → − → − 2. lim r (t) f (t) = Lr L f t→t0 3. lim → − →→ −− → f (t) · − (t) = L f · L g g 4. lim → − → − → − → f (t) × − (t) = L f × L g . g t→t0 t→t0 ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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