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III. Potensrækker – 1. l’Hospitals regel og uegentlige integraler 252 1.6. Overbevis ☞ [S] 4.5 Indeterminate forms and l’Hospital’s rule Bevis Fra den udvidede middelværdisætning Beregn nu f(x)g ′ (x ∗ ) = f ′ (x ∗ )g(x), a < x ∗ < x f(x) f lim = lim x→a g(x) x→a ′ (x∗ ) g ′ (x∗ f = lim ) x→a ′ (x) g ′ (x) 1.7. Prøv reglen ☞ [S] 4.5 Indeterminate forms and l’Hospital’s rule Eksempel ubestemt af form 0 0 . Heraf fås x lim x→1 2 − 1 x − 1 f(x) = x 2 − 1,f ′ (x) = 2x g(x) = x − 1,g ′ (x) = 1 x lim x→1 2 − 1 2x = lim = 2 x − 1 x→1 1
III. Potensrækker – 1. l’Hospitals regel og uegentlige integraler 253 1.8. l’Hospitals regel 2 ☞ [S] 4.5 Indeterminate forms and l’Hospital’s rule Sætning (l’Hospitals regel) Antag, at f,g er differentiable og g ′ (x) = 0 for x = a tilpas nœr a. Hvis f(x) lim x→a g(x) er et ubestemt udtryk af form ∞ ∞ , så er f(x) f lim = lim x→a g(x) x→a ′ (x) g ′ (x) 1.9. Prøv reglen ☞ [S] 4.5 Indeterminate forms and l’Hospital’s rule Eksempel (6) er ubestemt af form ∞ ∞ . Heraf fås lim x→0 ln |x| lim x→0 x−1 f(x) = ln |x|,f ′ (x) = |x| −1 g(x) = x −1 ,g ′ (x) = −x −2 ln |x| = lim x−1 x→0 |x| −1 x = lim −x = 0 −x−2 x→0 |x|
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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