●✐➳♦ ➳♥✿ ❚♦➳♥ ❝❛♦ ❝✃♣ ✷ ❱➢➡♥❣ ❚❤Þ ❚❤➯♦ ❇×♥❤ ➜❍ ◆❣♦➵✐ ❚❤➢➡♥❣ ❆♣r✐❧ ✶✽✱ ✷✵✵✻ ❝❤➢➡♥❣✶ ❇ætrî✈Ò♣❤Ð♣tÝ♥❤✈✐♣❤➞♥✈➭tÝ❝❤♣❤➞♥❤➭♠sè♠ét❜✐Õ♥sè §✶✳ ●✐í✐ ❤➵♥ ✈➭ sù ❧✐➟♥ tô❝ ❝ñ❛ ❤➭♠ sè✿ ✶✳ ●✐í✐ ❤➵♥ ❝ñ❛ ❤➭♠ sè ❛✳ ❈➳❝ ➤Þ♥❤ ♥❣❤Ü❛ • ➜◆ ✶✿ y = f(x) ①➳❝ ➤Þ♥❤ t➵✐ V (x 0 )✳ ✲ ●✐í✐ ❤➵♥ ❤÷✉ ❤➵♥ ❝ñ❛ ❤➭♠ sè a = lim x→x 0 f(x)    (x=x 0 ) ⇐⇒ ∀ > 0,∃β > 0 : 0 < |x − x 0 | < β    x=x 0 , x ➤ñ ❣➬♥ x 0 ⇒ |f(x) − a| <     f(x) ❣➬♥ a t✉ú ý ❑Ý ❤✐Ö✉ lim x→x 0 f(x) = a ❤♦➷❝ f(x) → a ❦❤✐ x → x 0 ❱Ý ❞ô ✶✿ ❉ï♥❣ ♥❣➠♥ ♥❣÷  − β✱ ❝❤ø♥❣ ♠✐♥❤ lim x→0 x cos 1 x = 0. ❈❤ø♥❣ ♠✐♥❤✿ ❍➭♠ f(x) ❦❤➠♥❣ ①➳❝ ➤Þ♥❤ t➵✐ x 0 = 0✱ ♥❤➢♥❣ ①➳❝ ➤Þ♥❤ t➵✐ ❧➞♥ ❝❐♥ x 0 ✱ ❝❤➻♥❣ ❤➵♥ t➵✐ V (x 0 ) = (−1, 1)✳ ▲✃②  > 0 ❜Ð t✉ú ý✳ ❳Ðt |f(x) − 0| ≤  ⇔ |x cos 1 x | = |x|| cos 1 x | ≤  ❈❤ä♥ β = ✱ t❛ ➤➢î❝ ∀ > 0,∃β =  > 0 : ∀x : 0 < |x − 0| < β ⇒ |f(x) − 0| <  ❱❐② lim x→0 x cos 1 x = 0. ✶ • ➜◆ ✷ ✭t❤❡♦ ♥❣➠♥ ♥❣÷  − β✮✿ ✲ ●✐í✐ ❤➵♥ ❤÷✉ ❤➵♥ ❝ñ❛ ❤➭♠ sè a = lim x→∞ f(x) ⇐⇒ ∀ > 0,∃∆ > 0 : |x| > ∆ ⇒ |f(x) − a| <  −∞ = lim x→x 0 f(x) ⇐⇒ ∀E > 0,∃δ > 0 : 0 < |x − x 0 | < δ ⇒ f(x) < −E −∞ = lim x→+∞ f(x) ⇐⇒ ∀E > 0,∃∆ > 0 : ∀x > ∆ ⇒ f(x) < −E +∞ = lim x→−∞ f(x) ⇐⇒ ∀E > 0,∃∆ > 0 : ∀x < −∆ ⇒ f(x) > E • ➜◆ ✸ ✭●✐í✐ ❤➵♥ ♠ét ♣❤Ý❛✮✿ ✲ ●✐í✐ ❤➵♥ ♣❤➯✐ ❝ñ❛ f(x) t➵✐ x 0 a = lim x→x + 0 f(x) ⇐⇒ ∀ > 0,∃β > 0 : 0 < x− x 0 < β ⇒ |f(x)− a| <  ❑Ý ❤✐Ö✉ f(x + 0 ) ❤♦➷❝ f(x 0 + 0)✳ ✲ ●✐í✐ ❤➵♥ tr➳✐ ❝ñ❛ f(x) t➵✐ x 0 a = lim x→x − 0 f(x) ⇐⇒ ∀ > 0,∃β > 0 : 0 < x 0 − x < β ⇒ |f(x) − a| <  ❑Ý ❤✐Ö✉ f(x − 0 ) ❤♦➷❝ f(x 0 − 0)✳ ❱Ý ❞ô ✸✿ ●✐í✐ ❤➵♥ ♣❤➯✐ lim x→5 + √ x − 5 = 0✳ ❑❤➠♥❣ tå♥ t➵✐ ❣✐í✐ ❤➵♥ tr➳✐ lim x→5 − √ x − 5✳ ❱Ý ❞ô ✹✿ ❈❤♦ f(x) = s❣♥x =      1, ♥Õ✉ x > 0 0, ♥Õ✉ x = 0 −1, ♥Õ✉ x < 0 ❚❛ ❝ã f(0 + ) = lim x→0+ f(x) = lim x→0+ 1 = 1 f(0 − ) = lim x→0− f(x) = lim x→0− −1 = −1 ✭●✐í✐ ❤➵♥ tr➳✐ ✈➭ ❣✐í✐ ❤➵♥ ♣❤➯✐ t➵✐ ✵ ❦❤➳❝ ♥❤❛✉✮✳ ❜✳ ➜✐Ò✉ ❦✐Ö♥ tå♥ t➵✐ ❣✐í✐ ❤➵♥ • ➜▲ ✶ ✭▼è✐ ❧✐➟♥ ❤Ö ❣✐÷❛ ❣✐í✐ ❤➵♥ ✈➭ ❣✐í✐ ❤➵♥ ♠ét ♣❤Ý❛✮✿ f(x) ❝ã ❣✐í✐ ❤➵♥ ❤÷✉ ❤➵♥ t➵✐ x 0 ⇐⇒  ∃f(x − 0 ),∃f(x + 0 ) f(x − 0 ) = f(x + 0 ) = lim x→x 0 f(x) ❚r♦♥❣ ✈Ý ❞ô tr➟♥✱ ❤➭♠ s❣♥x ❦❤➠♥❣ ❝ã ❣✐í✐ ❤➵♥ t➵✐ 0✳ ✷ • ➜▲ ✷✿ lim x→x 0 f(x) = a ⇐⇒ f(x) = a + α(x) tr♦♥❣ ➤ã α(x) ❧➭ ❱❈❇ ❦❤✐ x → x 0 ✭lim x→x 0 α(x) = 0✮ ❈❤ø♥❣ ♠✐♥❤✿ ❚õ ➤Þ♥❤ ♥❣❤Ü❛ ❣✐í✐ ❤➵♥✱ t❛ ❝ã ♥❣❛② ♥❤❐♥ ①Ðt s❛✉✿ lim x→x 0 f(x) = a ⇐⇒ lim x→x 0 [f(x) − a] = 0 ➜➷t f(x) − a = α(x)✱ t❛ ➤➢î❝ ➤✐Ò✉ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳ • ➜▲ ✸✿ ✭❚✐➟✉ ❝❤✉➮♥ ❣✐í✐ ❤➵♥ ❝❤♦ ❤➭♠ ➤➡♥ ➤✐Ö✉✮ ✶✳ ◆Õ✉ ❤➭♠ f(x) ①➳❝ ➤Þ♥❤✱ ➤➡♥ ➤✐Ö✉ t➝♥❣ ✭❣✐➯♠✮ tr➟♥ ♠✐Ò♥ [α, +∞) ✈➭ ❜Þ ❝❤➷♥ tr➟♥ ✭❞➢í✐✮✱ t❤× ∃ lim x→+∞ f(x)✳ ✷✳ ◆Õ✉ ❤➭♠ f(x) ①➳❝ ➤Þ♥❤✱ ➤➡♥ ➤✐Ö✉ t➝♥❣ ✭❣✐➯♠✮ tr➟♥ ♠✐Ò♥ (−∞, α] ✈➭ ❜Þ ❝❤➷♥ ❞➢í✐ ✭tr➟♥✮✱ t❤× ∃ lim x→−∞ f(x)✳ • ➜▲ ✹✿ ✭◆❣✉②➟♥ ❧ý ❦Ñ♣✮ ❈❤♦ ❤➭♠ f(x), u(x), v(x) ①➳❝ ➤Þ♥❤ ë ❧➞♥ ❝❐♥ U(x 0 )  u(x) ≤ f(x) ≤ v(x), ∀x ∈ U(x 0 ) lim x→x 0 u(x) = lim x→x 0 v(x) = a =⇒ ∃ lim x→x 0 f(x), lim x→x 0 f(x) = a. ✷✳ P❤Ð♣ t♦➳♥ ❣✐í✐ ❤➵♥ ❤➭♠ ❑Ý ❤✐Ö✉ x → x ∗ 0 ✿ x → x 0 ❤♦➷❝ x → ∞ ➜▲ ✺ ✭●✐í✐ ❤➵♥ ❝ñ❛ tæ♥❣✱ ❤✐Ö✉✱ tÝ❝❤✱ t❤➢➡♥❣✮✿ ❈❤♦ ❤➭♠ f(x) ✈➭ ❤➭♠ g(x) ❝ã ❣✐í✐ ❤➵♥ ❤÷✉ ❤➵♥ ❦❤✐ x → x ∗ 0 ✳ ❑❤✐ ➤ã✿ ✶✳ ∃ lim x→x ∗ 0 [f(x) ± g(x)] ✈➭ lim x→x ∗ 0 [f(x) ± g(x)] = lim x→x ∗ 0 f(x) ± lim x→x ∗ 0 g(x)] ✷✳ ∃ lim x→x ∗ 0 [f(x).g(x)] ✈➭ lim x→x ∗ 0 [f(x).g(x)] = lim x→x ∗ 0 f(x). lim x→x ∗ 0 g(x) ✸✳ ∃ lim x→x ∗ 0 f(x) g(x) ✈➭ ♥Õ✉ lim x→x ∗ 0 g(x) = 0 t❤× lim x→x ∗ 0 f(x) g(x) = lim x→x ∗ 0 f(x) lim x→x ∗ 0 g(x) ✸ ➜▲ ♥➭② ❝ã t❤Ó ♠ë ré♥❣ ❝❤♦ ❤÷✉ ❤➵♥ ❤➭♠ sè ❝ã ❣✐í✐ ❤➵♥ ❤÷✉ ❤➵♥ tr♦♥❣ ❝ï♥❣ ♠ét q✉➳ tr×♥❤ ♥➭♦ ➤ã✳ ➜▲ ✭●✐í✐ ❤➵♥ ❝ñ❛ ❤➭♠ ❤î♣✮✿ ❈❤♦ ❤➭♠ ❤î♣ f ◦ u : x → f[u(x)]✳ ◆Õ✉ ❛✳ lim x→x ∗ 0 u(x) = u 0 ❜✳ f(u) ①➳❝ ➤Þ♥❤ t➵✐ u 0 ✈➭ f(u) → f(u 0 ) ❦❤✐ u → u 0 t❤× lim x→x ∗ 0 f[u(x)] = f[ lim x→x ∗ 0 u(x)] = f(u 0 ) ✸✳ ▼ét sè ❝❤ó ý • ❑❤✐ tÝ♥❤ ❣✐í✐ ❤➵♥ t❛ ❝ã t❤Ó sö ❞ô♥❣ ❝➳❝ ❣✐í✐ ❤➵♥ ❝➡ ❜➯♥ s❛✉✿ ✶✮ lim x→0 sin x x = 1 ✷✮ lim x→∞  1 + 1 x  x = lim u→0 (1 + u) 1 u = e✳ ❍➭♠ s➡ ❝✃♣ ❝➡ ❜➯♥ s❛✉✿ ✰✮ ❍➭♠ ❧✉ü t❤õ❛ x α ✱ ❤➭♠ ♠ò α x ✱ ❤➭♠ ❧♦❣❛r✐t log a x ✰✮ ❈➳❝ ❤➭♠ ❧➢î♥❣ ❣✐➳❝✿ sin x, cos x, t❣x, ❝♦t❣x ✰✮ ✈➭ ❝➳❝ ❤➭♠ ❧➢î♥❣ ❣✐➳❝ ♥❣➢î❝✿ arcsin x, arccos x, ❛r❝t❣x, ❛r❝❝t❣x ❤➭♠ s➡ ❝✃♣ ❧➭ ❝➳❝ ❤➭♠ ➤➢î❝ t➵♦ t❤➭♥❤ ❜ë✐ ♠ét sè ❤÷✉ ❤➵♥ ❝➳❝ ♣❤Ð♣ tÝ♥❤ tæ♥❣✱ ❤✐Ö✉✱ tÝ❝❤✱ t❤➢➡♥❣ ✈➭ ❤î♣ ❤➭♠ ➤è✐ ✈í✐ ❤➭♠ ❤➺♥❣ ✈➭ ❝➳❝ ❤➭♠ s➡ ❝✃♣ ❝➡ ❜➯♥✳ • ◆Õ✉ ❤➭♠ sè s➡ ❝✃♣ f(x) ①➳❝ ➤Þ♥❤ ❦❤✐ x = a ✈➭ ❧➞♥ ❝❐♥ a t❤× lim x→a f(x) = f(a) tr♦♥❣ ➤ã • ◆Õ✉ f(x), g(x) ❝ã ❣✐í✐ ❤➵♥ ❤÷✉ ❤➵♥ ❦❤✐ x → x 0 ✈➭ lim x→x 0 f(x) > 0 t❤× lim x→x 0 f(x) g(x) =  lim x→x 0 f(x)  lim x→x 0 g(x) ✹✳ ❚Ý♥❤ ❝❤✃t ❝ñ❛ ❝➳❝ ❤➭♠ ❝ã ❣✐í✐ ❤➵♥ 0 ✭❱❈❇✮ ✈➭ ❝➳❝ ❤➭♠ ❝ã ❣✐í✐ ❤➵♥ ∞ ✭❱❈▲✮ ❳Ðt tr♦♥❣ ❝ï♥❣ ♠ét q✉➳ tr×♥❤✱ t❛ ❝ã ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉✿ ✶✳ ◆Õ✉ u → 0 t❤× 1 u → ∞ ✈➭ ♥❣➢î❝ ❧➵✐✱ ♥Õ✉ u → ∞ t❤× 1 u → 0✳ ✷✳ ❚æ♥❣✱ ❤✐Ö✉✱ tÝ❝❤ ❤÷✉ ❤➵♥ ❝➳❝ ❱❈❇ ❧➭ ❱❈❇ ✭①Ðt tr♦♥❣ ❝ï♥❣ ✶ q✉➳ tr×♥❤✮✳ ✹ ✸✳ ◆Õ✉ u → 0 ✈➭ v ❜Þ ❝❤➷♥ t❤× u.v → 0✳ ❱Ý ❞ô ✻✿ ❦❤✐ x → 0✱ ♠➷❝ ❞ï sin 1 x ❦❤➠♥❣ ❝ã ❣✐í✐ ❤➵♥✱ ❤➭♠ sè f(x) = x. sin 1 x ❝ã ❣✐í✐ ❤➵♥ ❧➭ 0 ✈× x ❧➭ ❱❈❇ ✈➭ | sin 1 x | ≤ 1 ✭❝ã t❤Ó ❝❤ø♥❣ ♠✐♥❤ ❜➺♥❣ ➤Þ♥❤ ♥❣❤Ü❛✮✳ ✹✳ ❚æ♥❣ u + v → ∞ tr♦♥❣ ❝➳❝ tr➢ê♥❣ ❤î♣ s❛✉✿ ✰ u → ∞ ✈➭ v → k ✭❤÷✉ ❤➵♥✮✳ ✰ u → ∞✱ v → ∞ ✈➭ u.v > 0✳ ✺✳ ❚Ý❝❤ uv → ∞ tr♦♥❣ ❝➳❝ tr➢ê♥❣ ❤î♣ s❛✉✿ ✰ u → ∞ ✈➭ v → ∞ ✳ ✰ u → ∞✱ v → k = 0✳ ✻✳ ◆Õ✉ f(x) ❧➭ ❱❈▲ tr♦♥❣ ♠ét q✉➳ tr×♥❤ ♥➭♦ ➤ã t❤× f(x) ❧➭ ❤➭♠ ❦❤➠♥❣ ❜Þ ❝❤➷♥ tr♦♥❣ q✉➳ tr×♥❤ ➤ã✳ ◆❣➢î❝ ❧➵✐✱ ❤➭♠ ❦❤➠♥❣ ❜Þ ❝❤➷♥ ❝❤➢❛ ❝❤➽❝ ❧➭ ❱❈▲ ✭✈Ý ❞ô f(x) = x cos x✮✳ ✼✳ ❚Ý❝❤ ❝ñ❛ ♠ét ❱❈▲ ✈í✐ ❤➭♠ ❜Þ ❝❤➷♥✱ ❝❤➢❛ ❝❤➽❝ ❧➭ ❱❈▲✳ ✺✳ ◗✉② t➽❝ t❤❛② t❤Õ ❱❈❇✱ ❱❈▲ t➢➡♥❣ ➤➢➡♥❣ ❚r♦♥❣ ♣❤➬♥ ♥➭②✱ t❛ ❦Ý ❤✐Ö✉ α(x), β(x) ❧➭ ❝➳❝ ❱❈❇ ✭❱❈▲✮ tr♦♥❣ ❝ï♥❣ ♠ét q✉➳ tr×♥❤✳ ✲ ◆Õ✉ lim α(x) β(x) = 1 t❤× t❛ ♥ã✐ α(x), β(x) ❧❛ ❤❛✐ ❱❈❇ ✭❱❈▲✮ t➢➡♥❣ ➤➢➡♥❣✱ ✈✐Õt α(x) ∼ β(x)✳ ✲ ❈❤♦ α(x), β(x), α(x), β(x) ❧➭ ❝➳❝ ❱❈❇ ✭❱❈▲✮ tr♦♥❣ ❝ï♥❣ ♠ét q✉➳ tr×♥❤✳ ❑❤✐ ➤ã ◆Õ✉  α(x) ∼ α(x) β(x) ∼ β(x) ⇒ lim α(x) β(x) = lim α(x) β(x) • ▼ét sè ❱❈❇ t➢➡♥❣ ➤➢➡♥❣ ❝➬♥ ♥❤í✿ ❑❤✐ x → 0 : sin x ∼ x arcsin x ∼ x t❣x ∼ x ❛r❝t❣x ∼ x 1 − cos x ∼ x 2 2 ln(1 + x) ∼ x e x − 1 ∼ x (1 − x) µ − 1 ∼ µx ❱Ý ❞ô✱ ❜➭✐ ✶✳✸✵✳✼ ✭tr ✶✻✮ lim x→0 π x ❝♦t❣ π 2 x = lim x→0 πt❣ π 2 x x = lim x→0 π π 2 x x = π 2 2 ❇➭✐ t❐♣✿ ❇➭✐ ✶✳✶✵✱ ❇➭✐ ✶✳✶✶✱ ❇➭✐ ✶✳✶✹✱ ❇➭✐ ✶✳✶✺✱ ❇➭✐ ✶✳✶✻ ✭❙➳❝❤ ❜➭✐ t❐♣✮✳ ✺ ✻✳ ❙ù ❧✐➟♥ tô❝ ❝ñ❛ ❤➭♠ sè ❛✳ ❈➳❝ ➤Þ♥❤ ♥❣❤Ü❛ • ▲✐➟♥ tô❝ t➵✐ ♠ét ➤✐Ó♠ x 0 ✿ f ∈ C(x 0 ) ⇐⇒ lim x→x 0 f(x) = f(x 0 ) ✭❱✐Õt ❣✐í✐ ❤➵♥ ♥❤➢ ✈❐② ❝ã ♥❣❤Ü❛ f(x) ①➳❝ ➤Þ♥❤ t➵✐ x 0 ✈➭ ❧➞♥ ❝❐♥ x 0 ✮✳ ✲ ▲✐➟♥ tô❝ ♣❤➯✐ t➵✐ x 0 ✿ f ∈ C(x + 0 ) ⇐⇒ lim x→x + 0 f(x) = f(x 0 ) ✲ ▲✐➟♥ tô❝ tr➳✐ t➵✐ x 0 ✿ f ∈ C(x − 0 ) ⇐⇒ lim x→x − 0 f(x) = f(x 0 ) ✲ ➜✐Ò✉ ❦✐Ö♥ ➤Ó f ❧✐➟♥ tô❝ t➵✐ x 0 f ∈ C(x 0 ) ⇐⇒  f ∈ C(x + 0 ) f ∈ C(x − 0 ) ⇐⇒      f(x) ①➳❝ ➤Þ♥❤ t➵✐ x 0 ✈➭ ❧➞♥ ❝❐♥ x 0 f(x + 0 ) = f(x 0 ) f(x − 0 ) = f(x 0 ) ✲ ▲✐➟♥ tô❝ tr♦♥❣ ❦❤♦➯♥❣ (a, b)✿ f(a, b) ∈ C ⇐⇒ f ∈ C(x), ∀x ∈ (a, b) ✲ ▲✐➟♥ tô❝ tr➟♥ ➤♦➵♥ [a, b]✿ f ∈ C[a, b] ⇐⇒      f ∈ C(a, b) f ∈ C(a + ) f ∈ C(b − ) ➜å t❤Þ ❤➭♠ ❧✐➟♥ tô❝ ❝ã t❤Ó ❣➲② ♥❤➢♥❣ ❦❤➠♥❣ ➤øt ✲ ❈❤ó ý✿ ❍➭♠ sè s➡ ❝✃♣ ❧✐➟♥ tô❝ t➵✐ ♠ä✐ ➤✐Ó♠ t❤✉é❝ ♠✐Ò♥ ①➳❝ ➤Þ♥❤ ❝ñ❛ ♥ã✳ ❱Ý ❞ô✿ ✻ ❛✮ ❚×♠ ♠✐Ò♥ ❧✐➟♥ tô❝ ❝ñ❛ ❝➳❝ ❤➭♠ s➡ ❝✃♣ s❛✉✿ y = x sin x x 2 − 1 x − 1 ❜✮ ❳➳❝ ➤Þ♥❤ a ➤Ó ❤➭♠ sè s❛✉ ❧✐➟♥ tô❝ tr➟♥ t♦➭♥ ❜é IR f(x) =  x 2 + a ❦❤✐ x ≥ 0 4 x ❦❤✐ x < 0 ●✐➯✐✿ ❛✮ ❍➭♠ s➡ ❝✃♣ ❧✐➟♥ tô❝ t➵✐ ♠ä✐ ➤✐Ó♠ t❤✉é❝ ♠✐Ò♥ ①➳❝ ➤Þ♥❤✱ ❞♦ ➤ã ❜➭✐ t♦➳♥ trë t❤➭♥❤ t×♠ ♠✐Ò♥ ①➳❝ ➤Þ♥❤ ❝ñ❛ ❤➭♠ sè s➡ ❝✃♣ y = x sin x ①➳❝ ➤Þ♥❤ ⇔ sin x = 0 ⇔ x = kπ, k ∈ ZZ x 2 −1 x−1 ①➳❝ ➤Þ♥❤ ⇔ x − 1 = 0 ⇔ x = 1 ❜✮ ✲ ❑❤✐ x < 0✱ f(x) = x 2 + a ❧➭ ❤➭♠ s➡ ❝✃♣ ♥➟♥ ❧✐➟♥ tô❝ tr➟♥ t♦➭♥ ❜é ♠✐Ò♥ ①➳❝ ➤Þ♥❤✳ ✲ ❑❤✐ x < 0✱ f(x) = 4 x ❧➭ ❤➭♠ s➡ ❝✃♣ ♥➟♥ ❧✐➟♥ tô❝ tr➟♥ t♦➭♥ ❜é ♠✐Ò♥ ①➳❝ ➤Þ♥❤✳ ✲ ❚❛ ①Ðt tÝ♥❤ ❧✐➟♥ tô❝ t➵✐ x = 0 f(0) = a f(0 + ) = lim x→0 + (x 2 + a) = 0 + a = a f(0 − ) = lim x→0 − (4 x ) = 4 0 = 1 f ❧✐➟♥ tô❝ t➵✐ x = 0 ⇐⇒ f(0 + ) = f(0 − ) = f(0) ⇐⇒ a = 1 ❱❐② f ❧✐➟♥ tô❝ tr➟♥ t♦➭♥ ❜é IR ❦❤✐ a = 1✳ • ➜✐Ó♠ ❣✐➳♥ ➤♦➵♥ ❍➭♠ f(x) ➤➢î❝ ❣ä✐ ❧➭ ❣✐➳♥ ➤♦➵♥ t➵✐ x 0 ✱ ♥Õ✉ ♥ã ❦❤➠♥❣ ❧✐➟♥ tô❝ t➵✐ x 0 ✳ ❑❤✐ ➤ã ➤✐Ó♠ x 0 ❣ä✐ ❧➭ ➤✐Ó♠ ❣✐➳♥ ➤♦➵♥ ❝ñ❛ ❤➭♠ f(x) ✈➭ ✈✐Õt f = C(x 0 )✳ ❱Ý ❞ô✿ ❈➳❝ ❤➭♠ y = 1 x − 1 y = e 1 x−1 ❣✐➳♥ ➤♦➵♥ t➵✐ x = 1 ✈× ❝❤ó♥❣ ❦❤➠♥❣ ①➳❝ ➤Þ♥❤ t➵✐ ➤✐Ó♠ ➤ã✳ • P❤➞♥ ❧♦➵✐ ➤✐Ó♠ ❣✐➳♥ ➤♦➵♥ ✰ ➜✐Ó♠ ❣✐➳♥ ➤♦➵♥ ❧♦➵✐ ✶✿ ◆Õ✉ x 0 ❧➭ ➤✐Ó♠ ❣✐➳♥ ➤♦➵♥ ❝ñ❛ ❤➭♠ f(x) ✈➭ ❝ã ❣✐í✐ ❤➵♥ ♣❤➯✐ f(x + 0 )✱ ❣✐í✐ ❤➵♥ tr➳✐ f(x − 0 ) ❤÷✉ ❤➵♥ t❤× t❛ ♥ã✐ x 0 ❧➭ ➤✐Ó♠ ❣✐➳♥ ➤♦➵♥ ❧♦➵✐ ✶ ❝ñ❛ f(x)✳ ✰ ➜✐Ó♠ ❣✐➳♥ ➤♦➵♥ ❧♦➵✐ ✷✿ ❈➳❝ ➤✐Ó♠ ❣✐➳♥ ➤♦➵♥ ❦❤➠♥❣ ♣❤➯✐ ❣✐➳♥ ➤♦➵♥ ❧♦➵✐ ✶ ❣ä✐ ❧➭ ❣✐➳♥ ➤♦➵♥ ❧♦➵✐ ✷ ✼ ✰ ➜✐Ó♠ ❣✐➳♥ ➤♦➵♥ ❦❤ö ➤➢î❝✿ ●✐➯ sö x 0 ❧➭ ➤✐Ó♠ ❣✐➳♥ ➤♦➵♥ ♠➭ f(x − 0 ) = f(x + 0 ) = f(x 0 ) ❦❤✐ ➤ã t❛ ♥ã✐ x 0 ❧➭ ➤✐Ó♠ ❣✐➳♥ ➤♦➵♥ ❦❤ö ➤➢î❝ ✭♥❣❤Ü❛ ❧➭ ❝ã t❤Ó sö❛ ❧➵✐ t❤➭♥❤ ➤✐Ó♠ ❧✐➟♥ tô❝✮✳ ❱Ý ❞ô✿ ❚×♠ ❝➳❝ ➤✐Ó♠ ❣✐➳♥ ➤♦➵♥ ❝ñ❛ ❤➭♠ s❛✉ ✈➭ ♣❤➞♥ ❧♦➵✐ ➤✐Ó♠ ❣✐➳♥ ➤♦➵♥✿ y = sin x |x| 1 1 − e 1−x ●✐➯✐✿ ❛✮ y = sin x |x| =  sin x x ♥Õ✉ x > 0 − sin x x ♥Õ✉ x < 0 ❍➭♠ sè ❦❤➠♥❣ ①➳❝ ➤Þ♥❤ t➵✐ x = 0 ♥➟♥ ❣✐➳♥ ➤♦➵♥ t➵✐ x = 0✳ ❚❛ ❝ã✿ f(0 + ) = lim x→0 + sin x x = 1 f(0 − ) = lim x→0 − − sin x x = −1 ❱× f(0 + ) = f(0 − ) ♥➟♥ x = 0 ❧➭ ➤✐Ó♠ ❣✐➳♥ ➤♦➵♥ ❧♦➵✐ ✶✱ ❦❤➠♥❣ ❦❤ö ➤➢î❝✳ ❜✮ ❍➭♠ sè ❦❤➠♥❣ ①➳❝ ➤Þ♥❤ t➵✐ x = 1 ♥➟♥ ❣✐➳♥ ➤♦➵♥ t➵✐ x = 1✳ ❚❛ ❝ã✿ lim x→1 1 1 − e 1−x = ∞ ♥➟♥ x = 1 ❧➭ ➤✐Ó♠ ❣✐➳♥ ➤♦➵♥ ❧♦➵✐ ✷✳ ❜✳ ❈➳❝ ♣❤Ð♣ tÝ♥❤ ➤➡♥ ❣✐➯♥ ✈Ò ❤➭♠ ❧✐➟♥ tô❝ ➜▲ ✶✿  f ∈ C(x 0 ) g ∈ C(x 0 ) =⇒      f ± g ∈ C(x 0 ) f.g ∈ C(x 0 ) f g ∈ C(x 0 )✭ ✈í✐ g(x) = 0✮ ➜▲ ✷✿ ✭❙ù ❧✐➟♥ tô❝ ❝ñ❛ ❤➭♠ ❤î♣✮  u(x) ∈ C(x 0 ) f(u) ∈ C(u 0 = u(x 0 )) =⇒ f[u(x)] ∈ C(x 0 ) ❍Ö q✉➯✿ ❍➭♠ s➡ ❝✃♣ f(x) ①➳❝ ➤Þ♥❤ t➵✐ x 0 ✈➭ ❧➞♥ ❝❐♥ x 0 t❤× ❧✐➟♥ tô❝ t➵✐ x 0 ❝✳ ❈➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ ❤➭♠ ❧✐➟♥ tô❝ tr➟♥ ♠ét ➤♦➵♥ ✽ ➜▲ ✶✿ ✭❱➞②❡rtr❛s✮ f(x) ∈ C[a, b] ⇒  ∃K > 0 : |f(x)| ≤ K,∀x ∈ [a, b] ∃c, c  ∈ [a b] : m = f(c  ) ≤ f(x) ≤ f(c) = M,∀x ∈ [a, b] ➜▲ ✷✿  f(x) ∈ C[a, b] f(a)f(b) < 0 =⇒ ∃c ∈ (a, b) : f(c) = 0 ✭c ❣ä✐ ❧➭ ❦❤➠♥❣ ➤✐Ó♠✮ ➜▲ ✸ ✭✈Ò ❣✐➳ trÞ tr✉♥❣ ❣✐❛♥✮✿  f(x) ∈ C[a, b] µ ∈ [m, M] =⇒ ∃c ∈ [a, b] : f(c) = µ ▼➠ t➯ ❤×♥❤ ❤ä❝✿ ➜➢ê♥❣ ❝♦♥❣ ❧✐Ò♥ tr➟♥ ♠ét ➤♦➵♥ ➤ã♥❣ ❧✉➠♥ ❝ã ➤➷❝ ➤✐Ó♠✿ ✰ ❜Þ ❦Ñ♣ ❣✐÷❛ ❝➳❝ ➤➢ê♥❣ t❤➻♥❣ y = ±K✱ ✰ ❝ã ❣✐➳ trÞ ❧í♥ ♥❤✃t✱ ❣✐➳ trÞ ❜Ð ♥❤✃t tr➟♥ ➤♦➵♥ ➤ã✱ ✰ ◆Õ✉ ➤➢ê♥❣ ❝♦♥❣ ❧✐Ò♥ tr➟♥ ♠ét ➤♦➵♥ ➤ã♥❣ ➤✐ tõ ❜➟♥ ♥➭② s❛♥❣ ❜➟♥ ❦✐❛ trô❝ ❤♦➭♥❤ t❤× sÏ ❝➽t trô❝ ❤♦➭♥❤ Ýt ♥❤✃t ✶ ❧➬♥✳ ✰ ♠ä✐ ➤➢ê♥❣ t❤➻♥❣ y = γ ✈í✐ γ ∈ [m, M] ❝➽t ➤➢ê♥❣ ❝♦♥❣ ❧✐Ò♥ Ýt ♥❤✃t ✶ ➤✐Ó♠✱ ❇➭✐ t❐♣ ❧✉②Ö♥ t❐♣✿ ❇➭✐ ✶✳✶✽✱ ❇➭✐ ✶✳✶✾✱ ❇➭✐ ✶✳✷✵ §✷✳ ➜➵♦ ❤➭♠ ✈➭ ✈✐ ♣❤➞♥ ✷✳✶✳ ➜➵♦ ❤➭♠ ✷✳✶✳✶✳ ❈➳❝ ➜◆ ➜◆ ✶✿ ➜➵♦ ❤➭♠ ❝ñ❛ ❤➭♠ sè✿ f(x) ①➳❝ ➤Þ♥❤ t➵✐ x 0 ✱ U(x 0 ) ✳ x 0 + ∆x ∈ U(x 0 )✳ ◆Õ✉ ∃ lim ∆x→0 ∆f ∆x = lim ∆x→0 f(x 0 + ∆x) − f(x 0 ) ∆x = A ✭❤÷✉ ❤➵♥✮ t❤× t❛ ♥ã✐ r➺♥❣ ❤➭♠ f(x) ❝ã ➤➵♦ ❤➭♠ t➵✐ x 0 ✈➭ A ❧➭ ➤➵♦ ❤➭♠ ❝ñ❛ f(x) t➵✐ x 0 ✾ f  (x 0 ), y  (x 0 ), df(x 0 ) dx , df dx     x 0 , dy dx     x 0 ✲ ý ♥❣❤Ü❛ ❤×♥❤ ❤ä❝✿ f  (x 0 ) ❧➭ ❤Ö sè ❣ã❝ ❝ñ❛ t✐Õ♣ t✉②Õ♥ M 0 T 0 t➵✐ ➤✐Ó♠ M 0 ❝ñ❛ ➤➢ê♥❣ ❝♦♥❣✳ ❱Ý ❞ô✿ ❚Ý♥❤ f  (0) ✈í✐✿ f(x) = x(x − 1)(x − 2) .(x − 1999) ●✐➯✐✿ ❚❛ ❝ã✿ f(0 + ∆x) = f(∆x) = ∆x(∆x − 1)(∆x − 2) .(∆x − 1999) f(0) = 0 lim ∆x→0 f(0 + ∆x) − f(0) ∆x = lim ∆x→0 (∆x − 1)(∆x − 2) .(∆x − 1999) = −1999! ❱❐② f  (0) = −1999! ➜◆ ✷✿ ➜➵♦ ❤➭♠ ♠ét ♣❤Ý❛ f(x) ①➳❝ ➤Þ♥❤ t➵✐ x 0 ✈➭ ❧➞♥ ❝❐♥ ♣❤➯✐ U(x 0 ) ❝ñ❛ x 0 ✳ x 0 + ∆x ∈ U(x 0 )✳ ◆Õ✉ ∃ lim ∆x→0 + ∆f ∆x = A ✭❤÷✉ ❤➵♥✮ t❤× A ➤➢î❝ ❣ä✐ ❧➭ ➤➵♦ ❤➭♠ ♣❤➯✐ ❝ñ❛ f(x) t➵✐ x 0 ❦Ý ❤✐Ö✉✿ f  (x + 0 ), y  (x + 0 ) ➜◆ ➤➵♦ ❤➭♠ tr➳✐ ❝ñ❛ f(x) t➵✐ x 0 ❧➭ f  (x − 0 ) = lim ∆x→0 − ∆f ∆x ❈❤ó ý✿ f  (x − 0 ), f  (x + 0 ) ❧➭ ❤Ö sè ❣ã❝ t✐Õ♣ t✉②Õ♥ tr➳✐✱ ♣❤➯✐ ✭t➢➡♥❣ ø♥❣✮ t➵✐ M 0 ❑Ý ❤✐Ö✉ sù tå♥ t➵✐ ➤➵♦ ❤➭♠ t➵✐ x 0 ✿ f ∈ C 1 (x 0 )✳ ❑Ý ❤✐Ö✉ sù tå♥ t➵✐ ➤➵♦ ❤➭♠ tr➳✐✱ ♣❤➯✐ t➵✐ x 0 ✿ f ∈ C 1 (x − 0 ), f ∈ C 1 (x + 0 )✳ ➜▲ ✶✿ ❈❤♦ f(x) ❧➭ ❤➭♠ ①➳❝ ➤Þ♥❤ t➵✐ x 0 ✈➭ U(x 0 )✳ ✶✵