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2.3 Basic Limit Laws

2.3 Basic Limit Laws. Theorem Basic Limit Laws : If the lim f(x) and x c lim g(x) exist, then x c Sum Law: lim (f(x)) + g(x)) exists and x c lim (f(x) +g(x)) = lim f(x) + lim g(x)

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2.3 Basic Limit Laws

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  1. 2.3 Basic Limit Laws • Theorem Basic Limit Laws : If the lim f(x) and x c lim g(x) exist, then x c Sum Law: lim (f(x)) + g(x)) exists and x c lim (f(x) +g(x)) = lim f(x) + lim g(x) x c x c x c

  2. Cont. • Constant Law: For any number k, lim k f(x) exists and x c lim k f(x) = k lim f(x) x c x c • Product Law: lim f(x) g(x) exists and x c lim f(x)g(x) = (lim f(x))(lim g(x)) x c x c x c

  3. Cont. • Quotient Law : If lim g(x) ≠0, then limf(x) exists and x c x c g(x) limf(x) = lim f(x) x c g(x) x c____ lim g(x) x c

  4. Cont. • Powers and Roots: If p,q are integers with q≠0, then lim [f(x)](p/q) exists x c lim [f(x) ](p/q)=(lim f(x))(p/q)) x c x c Assume that lim f(x) ≥0 if p q is even, that lim f(x) ≠0 if (p/q) <0. In particular, for n a positive integer, lim [f(x)]n=(lim f(x))n, limn√f(x) = n√limf(x) x c x c x c x c

  5. Cont. In the second limit, assume that lim f(x) ≥0 if n is even. • Ex. 1 • (a ) lim x3 (b) lim (x3+ 5x + 7) (c) lim √x3 +5x +7) x 2 x 2 x 2

  6. Ex. 2 Evaluate • (a) limt+6 and (b) lim t(-1/4)(t+5)(1/3) x 1 2t 4 x 3

  7. Ex. 3 Assumptions Matter • Show that the Product Law cannot be applied to lim f(x) g(x) if f(x)=x g(x) = x-1. x 0 Solution For all x≠0 we have f(x)g(x) =x∙x-1=1, so the limit of the product exists: However, lim x-1 approaches ∞ as x 0+ and it approaches it approaches -∞ as x 0-. Therefore, the Product Law cannot be applied and its conclusion does not hold: (lim f(x) ) (lim g(x) ) =(lim x)(lim x-1) x 0 x 0 x 0 x 0 DNE

  8. Conclusion • The Basic Limit Laws : If lim f(x) and lim g(x) both exist, then (i) lim ( f(x) + g(x) ) = lim f(x) + lim g(x) x c x c x c (ii) lim k f(x) = k lim f(x) x c x c (iii) lim f(x) g(x) = (lim f(x)) (lim g(x)) x c x c x c (iv) lim g(x) ≠0, then lim f(x) = lim f(x) x c x c x c_ lim g(x) x c (v) If p,q are integers with q≠0, lim[f(x) (p/q) = (lim f(x))(p/q)) x c x c For n a positive integer, lim [f(x)]n=(lim f(x))n , limn√f(x) = n√lim f(x) x c x c x c • If the lim f(x) or lim g(x) DNE, then the Basic Limit Laws cannot be applied x c x c

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