●✐➯✐ tÝ❝❤ ✶ ✭❉➭♥❤ ❝❤♦ ❝➳❝ ♥❣➭♥❤ ❙➢ ♣❤➵♠ ❚ù ♥❤✐➟♥✮ ❍➢í♥❣ ❞➱♥ ❧➭♠ ❜➭✐ ❦✐Ĩ♠ tr❛ ❈❤➢➡♥❣ ✷ ❈➞✉ ✶✳ ❚×♠ ♣❤➳t ❜✐Ĩ✉ s❛✐ tr♦♥❣ ❝➳❝ ♣❤➳t ❜✐Ĩ✉ s❛✉✿ ❛✮ ❍➭♠ sè f (x) = sin x ❧➭ ❤➭♠ sè ❧❰❀ ❜✮ ❍➭♠ sè f (x) = x2 ❝✮ ❍➭♠ sè f (x) = cos x ❧➭ ❤➭♠ sè ❝❤➼♥❀ ❞✮ ❍➭♠ sè f (x) = x2 + x ❧➭ ❤➭♠ sè ❧❰✳ ❈➞✉ ✷✳ f : R → R✳ ❈❤ä♥ ♠Ư♥❤ ➤Ị ➤ó♥❣ tr♦♥❣ ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉✿ ●✐➯ sö lim f (x) = −∞ ❛✮ x>A ❧➭ ❤➭♠ sè ❝❤➼♥❀ x→+∞ t❤× f (x) ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ✈í✐ ♠ä✐ M >0 ❧✉➠♥ tå♥ t➵✐ x→−∞ t❤× f (x) < M ❀ ❝✮ x>M x< lim f (x) = l ∈ R x→−∞ t❤× |f (x) ❞✮ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ✈í✐ ♠ä✐ M >0 x < −A s❛♦ ❝❤♦ ♥Õ✉ ❧✉➠♥ tå♥ t➵✐ ε>0 s❛♦ ❝❤♦ ♥Õ✉ − l| < ε❀ lim f (x) = l ∈ R ❦❤✐ x→−∞ −M t❤× |f (x) ❈➞✉ ✸✳ s❛♦ ❝❤♦ ♥Õ✉ < −M ❀ lim f (x) = −∞ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ✈í✐ ♠ä✐ A > ❧✉➠♥ tå♥ t➵✐ M ❜✮ A>0 ✈➭ ❝❤Ø ❦❤✐ ✈í✐ ♠ä✐ M > ❧✉➠♥ tå♥ t➵✐ ε > s❛♦ ❝❤♦ ♥Õ✉ − l| < ε✳ ❈❤ä♥ ♠Ư♥❤ ➤Ị s❛✐ tr♦♥❣ ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉✿ ❛✮ ◆Õ✉ ♠ét ❤➭♠ số ó tì trị ❞✉② ♥❤✃t❀ ❜✮ ◆Õ✉ ❤➭♠ {xn } ⊂ X ♠➭ ①➳❝ ➤Þ♥❤ tr➟♥ t❐♣ X✱ a ∈ X ✈➭ lim f (x) = l ∈ R✱ x→a x→a > g(x) ✈í✐ ♠ä✐ x ∈ X ✈➭ tå♥ t➵✐ lim f (x)✱ lim g(x)✱ t❤× lim f (x) > lim g(x)❀ f (x) > g(x) ✈í✐ ♠ä✐ x ∈ X x→a ✈➭ tå♥ t➵✐ lim g(x)✳ x→a ❈➞✉ ✹✳ ❈❤ä♥ ♠Ö♥❤ ➤Ị ❛✮ ◆Õ✉ ❜✮ ◆Õ✉ ❝✮ ◆Õ✉ ❞✮ ◆Õ✉ t❤× ✈í✐ ♠ä✐ ❞➲② xn → a ❦❤✐ n → ∞ t❛ ❝ã lim f (xn ) = l❀ ❝✮ ◆Õ✉ f (x) ❞✮ ◆Õ✉ f ➤ó♥❣ tr♦♥❣ ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉✿ lim f (x) = a✱ t❤× lim |f (x)| = |a|❀ x→x0 x→x0 lim |f (x)| = |a|✱ t❤× lim f (x) = a❀ x→x0 x→x0 lim |f (x)| = |a|✱ t❤× lim+ f (x) = a❀ x→x0 x→x0 lim |f (x)| = |a|✱ t❤× lim− f (x) = −a✳ x→x0 x→x0 ✶ x→a x→a lim f (x)✱ lim g(x)✱ x→a x→a x→a t❤× lim f (x) ≥ x→a ❈➞✉ ✺✳ ●✐➯ sö f, g : (a, b) → R✱ x0 ∈ (a, b)✳ ▼Ư♥❤ ➤Ị ♥➭♦ s❛✉ ➤➞② ➤ó♥❣✿ ❛✮ ◆Õ✉ f (x) > tr♦♥❣ ❧➞♥ ❝❐♥ U ❜✮ ◆Õ✉ f (x) ≥ g(x) tr♦♥❣ ❧➞♥ ❝❐♥ U ❝đ❛ ➤✐Ĩ♠ x0 ✈➭ f (x) > g(x) tr♦♥❣ ❧➞♥ ❝❐♥ U ❝đ❛ ➤✐Ĩ♠ x0 ✈➭ ❝đ❛ ➤✐Ĩ♠ x0 ✈➭ lim f (x) = a✱ t❤× a > 0❀ x→x0 lim f (x) = a✱ lim g(x) = b✱ t❤× x→x0 x→x0 a ≥ b❀ ❝✮ ◆Õ✉ lim f (x) = a✱ lim g(x) = b✱ x→x0 x→x0 t❤× a > b❀ ❞✮ ◆Õ✉ ❈➞✉ ✻✳ ❛✮ ❜✮ f (x) < tr♦♥❣ ❧➞♥ ❝❐♥ U ❈❤♦ lim x→x0 x0 ✈➭ lim f (x) = a✱ t❤× a < 0✳ x→x0 lim f (x) = L = 0✳ ❚r♦♥❣ ❝➳❝ ❦Õt q✉➯ s❛✉ ➤➞②✱ ❦Õt q✉➯ ♥➭♦ s❛✐❄ √ f (x) = L❀ x→x0 lim ❝ñ❛ ➤✐Ó♠ = ❀ f (x) L 2 ❝✮ lim (f (x)) = L ❀ x→x0 √ ❞✮ lim f (x) = L✳ x→x0 x→x0 ❈➞✉ ✼✳ ●✐➳ trÞ ❝đ❛ ❣✐í✐ ❤➵♥ ❛✮ L = 1❀ ❜✮ L = −1❀ ❝✮ L = e❀ ❞✮ L = e2 ✳ ❈➞✉ ✽✳ ❈❤ä♥ ♠Ư♥❤ ➤Ị s❛✐ L = lim x→∞ x2 + x2 − x ❧➭✿ tr♦♥❣ ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉✿ ❛✮ ❍➭♠ sè f (x) ❝ã ❣✐í✐ ❤➵♥ ❦❤✐ x → a✱ t❤× ❝ã ❣✐í✐ ❤➵♥ tr➳✐ ✈➭ ❣✐í✐ ❤➵♥ ♣❤➯✐ ❦❤✐ x → a✳ ❜✮ ❍➭♠ sè f (x) ❝ã ❣✐í✐ ❤➵♥ ❦❤✐ x → a✱ t❤× ❝ã ❣✐í✐ ❤➵♥ tr➳✐ ❦❤✐ x → a✳ ❝✮ ❍➭♠ sè ❝ã ❣✐í✐ ❤➵♥ tr➳✐ ✈➭ ❣✐í✐ ❤➵♥ ♣❤➯✐ ❦❤✐ x→a t❤× ❤➭♠ ➤ã ❝ã ❣✐í✐ ❤➵♥ ❦❤✐ x → a✳ ❞✮ ❍➭♠ sè ❈➞✉ ✾✳ ❛✮ ❜✮ ❝✮ ❞✮ ❈❤♦ f (x) ❝ã ❣✐í✐ ❤➵♥ ❦❤✐ x → a✱ t❤× ❝ã ❣✐í✐ ❤➵♥ ♣❤➯✐ ❦❤✐ x → a✳ lim f (x) = a✱ lim g(x) = b✱ a, b ∈ R✳ ❑Õt ❧✉❐♥ ♥➭♦ s❛✉ ➤➞② ❧➭ s❛✐✿ x→x0 f (x) lim = x→x0 a ❀ g(x) b lim [f (x).g(x)] = a.b❀ x→x0 x→x0 lim [f (x) − g(x)] = a − b❀ x→x0 lim [f (x) + kg(x)] = a + kb ∀k ∈ R✳ x→x0 ❈➞✉ ✶✵✳ ❈❤ä♥ ♠Ư♥❤ ➤Ị ❛✮ ❍➭♠ sè s❛✐ tr♦♥❣ ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉✿ f (x) ❝ã ❣✐í✐ ❤➵♥ ❦❤✐ x → a✱ t❤× ❝ã ❣✐í✐ ❤➵♥ tr➳✐ ✈➭ ❣✐í✐ ❤➵♥ ♣❤➯✐ ❦❤✐ x → a❀ ✷ ❦❤✐ ❜✮ ❍➭♠ sè f (x) = |x − 1| ❦❤➠♥❣ ❝ã ❣✐í✐ ❤➵♥ ♣❤➯✐ ❦❤✐ x → 1❀ ❝✮ ❍➭♠ sè |f | ❝ã ❣✐í✐ ❤➵♥ ❦❤✐ x → a ♥Õ✉ ❤➭♠ f ❞✮ ❍➭♠ sè f ❝ã ❣✐í✐ ❤➵♥ ❦❤✐ ❝ã ❣✐í✐ ❤➵♥ ❦❤✐ x → a❀ x → a ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ♥ã ❝ã ❣✐í✐ ❤➵♥ tr➳✐ ✈➭ ❣✐í✐ ❤➵♥ ♣❤➯✐ x → a ✈➭ ❤❛✐ ❣✐í✐ ❤➵♥ ♥➭② ❜➺♥❣ ♥❤❛✉ x → a✳ ❈➞✉ ✶✶✳ ●✐í✐ ❤➵♥ ❛✮ L = α❀ ❜✮ L = −α❀ ❝✮ L = 0❀ ❞✮ L = 1✳ ❈➞✉ ✶✷✳ (1 − x)α − = α ♥❤❐♥ ❣✐➳ trÞ ♥➭♦❄ x→0 x L = lim ❚×♠ ♠Ư♥❤ ➤Ị s❛✐❄ f (x) = x2 + 1, g(t) = tan t✱ t❤× (g◦ f )(x) = tan(x2 + 1)❀ ❛✮ ❱í✐ f (x) = sin x, g(t) = ln t✱ t❤× (g◦ f )(x) = ln(sin x)❀ √ √ ❝✮ ❱í✐ f (x) = sin x, g(t) = et ✱ t❤× (g◦ f )(x) = e sin x ❀ ❜✮ ❱í✐ f (x) = cos x, g(t) = ln t✱ t❤× (g◦ f )(x) = cos(ln(t))✳ ❞✮ ❱í✐ ❈➞✉ ✶✸✳ ❈❤ä♥ ♠Ư♥❤ ➤Ị s❛✐ tr♦♥❣ ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉✿ ❛✮ ❑❤✐ x → t❛ ❝ã sin x ∼ arcsin x❀ ❜✮ ❑❤✐ x → t❛ ❝ã tan 3x ∼ sin 3x❀ ❝✮ ❑❤✐ x → t❛ ❝ã tan x ∼ cos x❀ ❞✮ ❑❤✐ x → t❛ ❝ã (e2x − 1) ∼ tan x✳ ❈➞✉ ✶✹✳ ❈❤♦ ❤➭♠ sè f (x) = 1 + e x−1 ✳ ì ị s tr ị s ❛✮ ❜✮ ❝✮ ❞✮ lim f (x) = 2❀ x→1− lim f (x) = 0❀ x→1+ lim f (x) = 2❀ x→1 lim f (x) = ✈➭ lim+ f (x) = 0✳ x→1− ❈➞✉ ✶✺✳ x→1 ❈❤ä♥ ♣❤➳t ❜✐Ó✉ ➤ó♥❣✳ ❚❛ ♥ã✐ f (x) ✈➭ ❱❈❇ ❜❐❝ ❝❛♦ ❤➡♥ s♦ ✈í✐ g(x) ❦❤✐ x → a ♥Õ✉✿ ❛✮ f (x) ✈➭ g(x) ❧➭ ❝➳❝ ❱❈❇ ❦❤✐ x → a ✈➭ lim ❜✮ f (x) ✈➭ g(x) ❧➭ ❝➳❝ ❱❈❇ ❦❤✐ x → a ✈➭ lim ❝✮ lim x→a x→a x→a f (x) g(x) = 0❀ ✸ f (x) g(x) f (x) g(x) = k ∈ R❀ = 1❀ ❞✮ f (x) ✈➭ g(x) ❧➭ ❝➳❝ ❱❈❇ ❦❤✐ x → a ✈➭ lim x→a f (x) g(x) = 0✳ f (x) ✈➭ g(x) ❧➭ ❝➳❝ ❱❈❇ t➢➡♥❣ ➤➢➡♥❣ ❦❤✐ x → a ♥Õ✉✿ f (x) ❛✮ f (x) ✈➭ g(x) ❧➭ ❝➳❝ ❱❈❇ ❦❤✐ x → a ✈➭ lim = 0❀ x→a g(x) f (x) ❜✮ f (x) ✈➭ g(x) ❧➭ ❝➳❝ ❱❈❇ ❦❤✐ x → a ✈➭ lim = 1❀ x→a g(x) f (x) = 0❀ ❝✮ lim x→a g(x) f (x) ❞✮ f (x) ✈➭ g(x) ❧➭ ❝➳❝ ❱❈❇ ❦❤✐ x → a ✈➭ lim = k✳ x→a g(x) ❈➞✉ ✶✻✳ ❚❛ ♥ã✐ ❈➞✉ ✶✼✳ ❑❤✐ x6 ❛✮ − ❜✮ x3 ❀ x2 ❝✮ ❞✮ x → 0✱ t❤× ln[cos(x3 )] t➢➡♥❣ ➤➢➡♥❣ ✈í✐ ❜✐Ĩ✉ t❤ø❝ ♥➭♦ s❛✉ ➤➞②✳ − ❀ ❀ x2 ✳ α > 0, α = 1✳ ❑Õt qñ❛ ♥➭♦ s❛✉ ➤➞② ➤ó♥❣❄ (1 + x)α − ❛✮ lim = 1❀ x→0 x (1 + x)α − = α❀ ❜✮ lim x→0 x (1 + x)α − ❝✮ ❑❤➠♥❣ tå♥ t➵✐ lim ❀ x→0 x (1 + x)α − = 0✳ ❞✮ lim x→0 x ❈➞✉ ✶✽✳ ❈➞✉ ✶✾✳ ❛✮ ●Ø❛ sö ➜➷t L = lim x→0 L = 1❀ x5 − 6x3 − 4x x4 − sin x ✳ ❑❤✐ ➤ã ❣✐➳ trÞ ❝đ❛ L ❧➭✿ L = −1❀ ❝✮ L = ❀ ❞✮ L = − ✳ ❜✮ ❈➞✉ ✷✵✳ ➜➷t L = lim x→0 ln(cos 2x) x2 ✳ ❑❤✐ ➤ã ❣✐➳ trÞ ❝đ❛ ✹ L ❧➭✿ ❛✮ L = 1❀ ❜✮ L = 2❀ ❝✮ L = −2❀ ❞✮ L = −1✳ ❈➞✉ ✷✶✳ ❛✮ 0❀ ❜✮ 1❀ ❝✮ 4❀ ❞✮ 2✳ ❈➞✉ ✷✷✳ ✈í✐ ●✐í✐ ❤➵♥ sin 8x − sin 4x lim sin 2x x→0 ❚×♠ ♠Ư♥❤ ➤Ị ♥❤❐♥ ❣✐➳ ♥➭♦❄ ➤ó♥❣✳ ❛✮ ◆Õ✉ f (x) ❧➭ ❱❈❇ ❦❤✐ x → a✱ t❤× ❧✐➟♥ tơ❝ t➵✐ x = a❀ ❜✮ ◆Õ✉ f (x), g(x) ❧➭ ❝➳❝ ❱❈❇ t➢➡♥❣ ➤➢➡♥❣ ❦❤✐ x → a ✈➭ f (x) ❧➭ ❱❈❇ ❜❐❝ ❝❛♦ ❤➡♥ s♦ h(x)✱ t❤× f (x)h(x) t➢➡♥❣ ➤➢➡♥❣ ✈í✐ g(x) ❦❤✐ x → a❀ ❝✮ ◆Õ✉ f (x) ❧✐➟♥ tơ❝ t➵✐ x = a ✈➭ ❜Þ ❝❤➷♥ tr♦♥❣ ❧➞♥ ❝❐♥ ❝đ❛ x = a✱ t❤× f (x) ❧➭ ❱❈❇ ❦❤✐ x → a❀ ❞✮ ◆Õ✉ t❤× f (x) ❧➭ ❱❈❇ ❜❐❝ ❝❛♦ ❤➡♥ s♦ ✈í✐ g(x) ✈➭ h(x) t➢➡♥❣ ➤➢➡♥❣ ✈í✐ f (x) ❦❤✐ x → a✱ h(x) ❧➭ ❱❈❇ ❜❐❝ ❝❛♦ ❤➡♥ s♦ ✈í✐ g(x) ❦❤✐ x → a✳ ❈➞✉ ✷✸✳ ➜➷t ❛✮ L = 1❀ ❜✮ L = 0❀ L = lim+ x→5 1 + e x−5 ✳ ❑❤✐ ➤ã ❣✐➳ trÞ ❝đ❛ L ❧➭✿ L = +∞❀ ❞✮ L = ✳ e ❝✮ ❈➞✉ ✷✹✳ ➜➷t ❛✮ L = 1❀ ❜✮ L = 0❀ L = lim− x→0 1 + ex ✳ ❑❤✐ ➤ã ❣✐➳ trÞ ❝đ❛ L = +∞❀ ❞✮ L = ✳ e ❝✮ ❈➞✉ ✷✺✳ ❛✮ ●✐í✐ ❤➵♥ lim x→0 e2x − x(x + 2) ♥❤❐♥ ❣✐➳ trÞ ♥➭♦❄ L = 1❀ ✺ L ❧➭✿ ❜✮ L = 2❀ ❝✮ L = 3❀ ❞✮ L = 4✳ 1− ❈➞✉ ✷✻✳ ●✐í✐ ❤➵♥ lim x→0 1+ ❛✮ L = 1❀ ❜✮ L = −1❀ ❝✮ L = +∞❀ ❞✮ L = −∞✳ ❈➞✉ ✷✼✳ x ♥❤❐♥ ❣✐➳ trÞ ♥➭♦❄ x ●✐➳ trÞ ❝đ❛ ❣✐í✐ ❤➵♥ lim (1 + 6x) 6x ❧➭✿ x→0 ❛✮ L= ❞✮ L = e✳ ❀ e2 ❜✮ L = 1❀ √ e❀ ❝✮ L = ❈➞✉ ✷✽✳ ❈❤ä♥ ♠Ö♥❤ ➤Ị s❛✐ tr♦♥❣ ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ ➤➞②✿ ❛✮ ◆Õ✉ ❤➭♠ f ❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥ ❦❤♦➯♥❣ ❜✮ ◆Õ✉ ❤➭♠ f ❧✐➟♥ tô❝ tr➟♥ ➤♦➵♥ ❝✮ ◆Õ✉ ❤➭♠ f ❧✐➟♥ tơ❝ tr➟♥ ➤♦➵♥ (a, b)✱ t❤× f [a, b]✱ t❤× f [a, b]✱ t❤× f ❧✐➟♥ tơ❝ tr➟♥ ❦❤♦➯♥❣ ➤ã❀ ❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥ ➤♦➵♥ ➤ã❀ ➤➵t ❣✐➳ trÞ ❧í♥ ♥❤✃t ✈➭ ❣✐➳ trÞ ❜Ð ♥❤✃t tr➟♥ ➤♦➵♥ ➤ã❀ f ❞✮ ◆Õ✉ ❤➭♠ ❈➞✉ ✷✾✳ ❧✐➟♥ tô❝ tr➟♥ ❦❤♦➯♥❣ ❈❤♦ f (a, b) tì f ị tr ➤➵t ❣✐➳ trÞ ❧í♥ ♥❤✃t tr➟♥ ❦❤♦➯♥❣ ➤ã✳ [a, b]✳ ❈❤ä♥ ♠Ư♥❤ ➤Ị s❛✐ tr♦♥❣ ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ ➤➞②✿ ❛✮ ◆Õ✉ ❤➭♠ f (a, b) ✈➭ lim+ f (x) = f (a)✱ lim− f (x) = f (b)✱ t❤× f x→a x→b [a, b]❀ ❧✐➟♥ tô❝ tr➟♥ ➤♦➵♥ ❜✮ ◆Õ✉ ❤➭♠ f ❧✐➟♥ tơ❝ tr➟♥ (a, b]✱ t❤× lim− f (x) = f (b)❀ ❝✮ ◆Õ✉ ❤➭♠ f ❧✐➟♥ tơ❝ tr➟♥ [a, b)✱ t❤× lim+ f (x) = f (a)❀ ❞✮ ◆Õ✉ ❤➭♠ f ❧✐➟♥ tô❝ tr➟♥ [a, b]✱ t❤× lim− f (x) = f (a)✱ lim+ f (x) = f (b)✳ ❈➞✉ ✸✵✳ ❍➭♠ ❧✐➟♥ tô❝ tr➟♥ ❦❤♦➯♥❣ f ❛✮ ●✐➯ sö x→b x→a x→b X ⊂ R✱ f : X → R✳ ❈❤ä♥ ♣❤➳t ❜✐Ĩ✉ ➤ó♥❣ tr♦♥❣ ❝➳❝ ♣❤➳t ❜✐Ĩ✉ s❛✉ ➤➞②✿ ➤➢ỵ❝ ❣ä✐ ❧➭ ❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥ f x→a X ♥Õ✉✿ ❧✐➟♥ tơ❝ ✈➭ ➤➵t ợ trị ỏ t t tr ✻ X❀ ε > 0✱ tå♥ t➵✐ δ > s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X ♠➭ |x − y| < ε✱ t❤× ε > 0✱ tå♥ t➵✐ δ > s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X ♠➭ |x − y| < δ ✱ t❤× |x − y| < δ ✱ t❤× ❜✮ ❱í✐ ♠ä✐ |f (x) − f (y)| < δ ❀ ❝✮ ❱í✐ ♠ä✐ |f (x) − f (y)| < ε❀ ε > 0✱ ❞✮ ❚å♥ t➵✐ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ δ > ✈➭ ✈í✐ ♠ä✐ x, y ∈ X ♠➭ |f (x) − f (y)| < ε✳ sin 2x ❈➞✉ ✸✶✳ ❈❤♦ ❤➭♠ sè f (x) = x 2k ❛✮ k = 0❀ ❜✮ k = 1❀ ❝✮ k=− ❞✮ k = 3✳ ❈➞✉ ✸✷✳ x>0 ♥Õ✉ x ≤ ❧✐➟♥ tơ❝ tr➟♥ R t❤× ❀ ❚×♠ ♠Ư♥❤ ➤Ị s❛✐ tr♦♥❣ ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ f (x) = x + sin x ❧✐➟♥ tô❝ tr➟♥ R❀ ❜✮ ❍➭♠ sè f (x) = x2 ❝✮ ❍➭♠ sè ❧✐➟♥ tô❝ tr➟♥ R❀ f (x) = sin x2 ❧✐➟♥ tô❝ tr➟♥ R❀ x2 ❍➭♠ sè f (x) = ❦❤➠♥❣ ❜Þ ❝❤➷♥ tr➟♥ R✳ x2 + ❈➞✉ ✸✸✳ ❚×♠ ♠Ư♥❤ ➤Ị s❛✐ tr♦♥❣ ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ f (x) ❧✐➟♥ tơ❝ tr➟♥ R✱ t❤× ❤➭♠ |f | ❝ị♥❣ ❧✐➟♥ tơ❝ tr➟♥ R❀ ❛✮ ◆Õ✉ ❤➭♠ sè ❜✮ f (x) = ln(x2 + 1) ❧✐➟♥ tô❝ tr➟♥ R❀ √ ❍➭♠ sè f (x) = x2 + ❧✐➟♥ tô❝ tr➟♥ R❀ √ ❍➭♠ sè f (x) = x2 − ❧✐➟♥ tơ❝ tr➟♥ R✳ ❞✮ f ❍➭♠ sè ❝✮ ➜Ĩ ❤➭♠ sè ❛✮ ❞✮ ❍➭♠ sè ❈➞✉ ✸✹✳ t➵✐ ♥Õ✉ ❚×♠ ♠Ư♥❤ ➤Ị g s❛✐ ❛✮ ◆Õ✉ f ✈➭ ❜✮ ◆Õ✉ f ❧✐➟♥ tô❝ t➵✐ ❝✮ ◆Õ✉ f.g ❞✮ ◆Õ✉ f tr♦♥❣ ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ ✷ ❤➭♠ sè ❧✐➟♥ tơ❝ t➵✐ a ✈➭ g a✱ t❤× ❤➭♠ h = 2f + 3g ❦❤➠♥❣ ❧✐➟♥ tô❝ t➵✐ a✱ t❤× ❤➭♠ ❝ị♥❣ ❧✐➟♥ tơ❝ t➵✐ h = f − 5g a❀ ❦❤➠♥❣ ❧✐➟♥ tơ❝ a❀ ❝đ❛ ❤➭♠ sè ❈➞✉ ✸✺✳ ❛✮ f ❧✐➟♥ tô❝ t➵✐ ✈➭ g a ✈➭ g ❧✐➟♥ tơ❝ t➵✐ a✱ t❤× ❤➭♠ f ❧➭ ✷ ❤➭♠ sè ❦❤➠♥❣ ❧✐➟♥ tô❝ t➵✐ h=f +g ❈❤♦ ❤➭♠ t➵✐ ❝ị♥❣ ❧✐➟♥ tơ❝ t➵✐ a❀ a✱ t❤× ❝❤➢❛ ❦Õt ợ tí tụ a f ị tr (c, d) ✈➭ a ∈ (c, d)✳ ❚×♠ ♠Ư♥❤ ➤Ị ➤ó♥❣❄ ❧✐➟♥ tơ❝ t➵✐ a✱ ♥Õ✉ lim+ f (x) = lim− f (x)❀ x→a x→a ✼ ❜✮ f ❧✐➟♥ tô❝ t➵✐ a✱ ♥Õ✉ lim+ f (x) = f (a)❀ ❝✮ f ❧✐➟♥ tô❝ t➵✐ a✱ ♥Õ✉ lim− f (x) = f (a)❀ ❞✮ f ❧✐➟♥ tô❝ t➵✐ a✱ ♥Õ✉ lim f (x) = f (a)✳ ❈➞✉ x→a x→a x→a sin x ✸✻✳ ❈❤♦ ❤➭♠ sè f (x) = x ❍➭♠ sè ❧✐➟♥ tô❝ t➵✐ ➤✐Ó♠ ♥Õ✉ x=0 ♥Õ✉ x = ❝✮ ❍➭♠ sè ❧✐➟♥ tơ❝ tr➳✐ t➵✐ ➤✐Ĩ♠ x = 0❀ x → 0✳ x ❈➞✉ ✸✼✳ ❈❤♦ ❤➭♠ sè f (x) = x2 + ax − ❜✮ a = 2❀ ❝✮ a = −1❀ ❞✮ a = 0✳ ❈➞✉ ✸✽✳ ❈❤♦ ❤➭♠ sè ❛✮ ❍➭♠ sè f ♥Õ✉ x≤1 ♥Õ✉ x > ➜Ĩ ❤➭♠ sè ❧✐➟♥ tơ❝ tr➟♥ ❝✮ P❤➢➡♥❣ tr×♥❤ f (x) = ❝ã Ýt ♥❤✃t ✷ ♥❣❤✐Ư♠ ✈í✐ ♠ä✐ m > 0❀ R t❤× m > s❛♦ ❝❤♦ ♣❤➢➡♥❣ tr×♥❤ f (x) = ✈➠ ♥❣❤✐Ư♠✳ ●✐➯ sư X ⊂ R✱ f : X → R✳ ❈❤ä♥ ♠Ư♥❤ ➤Ị ➤ó♥❣ tr♦♥❣ ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ ➤➞②✿ ❛✮ ◆Õ✉ f ❧✐➟♥ tô❝ tr➟♥ ❜✮ ◆Õ✉ f tụ ị tì ế f ❧✐➟♥ tô❝ tr➟♥ t❐♣ ❞✮ ◆Õ✉ f ❧✐➟♥ tô❝ tr➟♥ ❈➞✉ ✹✵✳ ❧✐➟♥ tô❝ tr➟♥ R❀ f (x) = ❝ã ✷ ♥❣❤✐Ư♠ ✈í✐ m = 0❀ ❈➞✉ ✸✾✳ f f (x) = x4 + mx2 − 1✳ ❑❤➻♥❣ ➤Þ♥❤ ♥➭♦ s❛✉ ➤➞② ❧➭ s❛✐❄ ❜✮ P❤➢➡♥❣ tr×♥❤ ❞✮ ❚å♥ t➵✐ ❛✮ x = 0❀ ❞✮ ❍➭♠ sè ❦❤➠♥❣ ❝ã ❣✐í✐ ❤➵♥ ❦❤✐ a = 1❀ s❛✐❄ x = 0❀ ❜✮ ❍➭♠ sè ❧✐➟♥ tơ❝ ♣❤➯✐ t➵✐ ➤✐Ĩ♠ ❛✮ ❑❤➻♥❣ ➤Þ♥❤ ♥➭♦ s❛✉ ➤➞② ❧➭ ❈❤♦ ❤➭♠ sè f X ✱ t❤× ❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥ X ❀ X ✱ t❤× f [a, b]✱ t❤× f lim f (x) = +∞❀ x→+∞ ❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥ t❐♣ ➤ã❀ ❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥ ❧✐➟♥ tơ❝ tr➟♥ [a, b]✳ [a, b]✳ ➜➷t f (a) = A, f (b) = B ✳ ❚×♠ ❦❤➻♥❣ ➤Þ♥❤ s❛✐ tr♦♥❣ ❝➳❝ ❦❤➻♥❣ ➤Þ♥❤ s❛✉ ➤➞②✿ ❛✮ ế f t t tì ợ x = f −1 (y) tå♥ t➵✐ ❧✐➟♥ tô❝ ✈➭ t➝♥❣ ♥❣➷t tr➟♥ [A, B] ế f t t tì ợ x = f −1 (y) tå♥ t➵✐ ❧✐➟♥ tô❝ ✈➭ t➝♥❣ ♥❣➷t tr➟♥ (A, B)❀ ❝✮ ◆Õ✉ f t➝♥❣ ♥❣➷t tì ợ x = f (y) tồ t ❧✐➟♥ tô❝ ✈➭ t➝♥❣ ♥❣➷t tr➟♥ (A, B]❀ ❞✮ ◆Õ✉ f t t tì ợ x = f (y) tå♥ t➵✐ ❧✐➟♥ tô❝ ✈➭ ❣✐➯♠ ♥❣➷t tr➟♥ [A, B]✳ ✽ ❈❤♦ ❤➭♠ sè ❈➞✉ ✹✶✳ ❦❤➻♥❣ ➤Þ♥❤ ➤ó♥❣ ❛✮ ◆Õ✉ f ❧✐➟♥ tô❝ tr➟♥ (a, b)✳ ➜➷t A = inf f (x), B = sup f (x)✳ x∈(a,b) ì x(a,b) tr ị s f t tì ợ x = f (y) tồ t➵✐ ❧✐➟♥ tô❝ ✈➭ ❣✐➯♠ ♥❣➷t tr➟♥ f ❣✐➯♠ ♥❣➷t tì ợ x = f (y) tồ t ❧✐➟♥ tô❝ ✈➭ ❣✐➯♠ ♥❣➷t tr➟♥ [A, B]❀ ❜✮ ◆Õ✉ (A, B)❀ ❝✮ ◆Õ✉ f ❞✮ ◆Õ✉ ❣✐➯♠ ♥❣➷t t❤× ❤➭♠ ♥❣➢ỵ❝ f x = f −1 (y) tå♥ t➵✐ ❧✐➟♥ tô❝ ✈➭ t➝♥❣ ♥❣➷t tr➟♥ (A, B]❀ t➝♥❣ ♥❣➷t tì ợ x = f (y) tồ t ❧✐➟♥ tô❝ ✈➭ ❣✐➯♠ ♥❣➷t tr➟♥ [A, B)✳ ❈➞✉ ✹✷✳ ❧➭ a, b ∈ R ✈➭ f : R → R ❧➭ ❤➭♠ ❧✐➟♥ tơ❝ tr➟♥ R✳ ❑❤➻♥❣ ➤Þ♥❤ ♥➭♦ s❛✉ ➤➞② s❛✐❄ [a, b] ⊂ R❀ ❛✮ f ❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥ ♠ä✐ ➤♦➵♥ ❜✮ f ❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥ ♠ä✐ ❦❤♦➯♥❣ ❝✮ f ❜Þ ❝❤➷♥ tr➟♥ ♠ä✐ ➤♦➵♥ ❞✮ f ❜Þ ❝❤➷♥ tr➟♥ ❈➞✉ ✹✸✳ ❧➭ ●✐➯ sư ●✐➯ sö (a, b) ⊂ R❀ [a, b] ⊂ R❀ R✳ a, b ∈ R ✈➭ f : R → R ❧➭ ❤➭♠ ❧✐➟♥ tơ❝ tr➟♥ R✳ ❑❤➻♥❣ ➤Þ♥❤ ♥➭♦ s❛✉ ➤➞② s❛✐❄ [a, b] ⊂ R❀ ❛✮ f ❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥ ♠ä✐ ➤♦➵♥ ❜✮ f ❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥ ♠ä✐ ❦❤♦➯♥❣ ❝✮ f ❜Þ ❝❤➷♥ tr➟♥ ♠ä✐ ➤♦➵♥ ❞✮ f ❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥ ❈➞✉ ✹✹✳ ❈❤♦ ❦❤➻♥❣ ➤Þ♥❤ ❛✮ ❜✮ ❝✮ (a, b) ⊂ R❀ [a, b] ⊂ R❀ R✳ f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 , n = 2k + 1, k ∈ N, an > 0✳ ❚×♠ ➤ó♥❣❄ P❤➢➡♥❣ tr×♥❤ f (x) = ❧✉➠♥ ❝ã ♥❣❤✐Ö♠ t❤ù❝❀ lim f (x) = −∞❀ x→+∞ lim f (x) = +∞❀ x→−∞ ❞✮ P❤➢➡♥❣ tr×♥❤ ❈➞✉ ✹✺✳ f (x) = ✈➠ ♥❣❤✐Ư♠✳ ❚×♠ ♠Ư♥❤ ➤Ị s❛✐ tr♦♥❣ ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉✿ ❛✮ ❍➭♠ sè ❧✐➟♥ tụ ị tr R tì lim |f (x)| = +∞ ❤♦➷❝ lim |f (x)| = x→+∞ x→−∞ +∞❀ ❜✮ ❍➭♠ sè ❧✐➟♥ tơ❝ tr➟♥ (a, b)✱ t❤× ➤➵t ❣✐➳ trÞ ❧í♥ ♥❤✃t tr➟♥ ❦❤♦➯♥❣ ➤ã❀ ❝✮ ❍➭♠ số tụ tr [a, b] tì t trị ❧í♥ ♥❤✃t ✈➭ ♥❤á ♥❤✃t tr➟♥ ➤♦➵♥ ➤ã❀ ❞✮ ❍➭♠ sè ❧✐➟♥ tơ❝ ✈➭ ➤➡♥ ➤✐Ư✉ t➝♥❣ ♥❣➷t tr➟♥ ✾ [a, b]✱ t❤× f (a) < f (x) < f (b) ✈í✐ ♠ä✐ x ∈ (a, b)✳ ❈➞✉ ✹✻✳ ●✐➯ sư f :R→R ❧➭ ❤➭♠ ❧✐➟♥ tơ❝ tr➟♥ ♥❣❤✐Ư♠ ♣❤➞♥ ❜✐Ưt ✈➭ ♣❤➢➡♥❣ tr×♥❤ f (x) = −c R✱ c = 0✱ ♣❤➢➡♥❣ tr×♥❤ f (x) = c ❝ã ✷ ❝ã ✶ ♥❣❤✐Ư♠✳ ❑❤➻♥❣ ➤Þ♥❤ ♥➭♦ s❛✉ ➤➞② ❧➭ ➤ó♥❣❄ ❛✮ P❤➢➡♥❣ tr×♥❤ f (x) = ✈➠ ♥❣❤✐Ư♠ ❀ ❜✮ P❤➢➡♥❣ tr×♥❤ f (x) = ❝ã Ýt ♥❤✃t ✶ ♥❣❤✐Ư♠❀ ❝✮ P❤➢➡♥❣ tr×♥❤ f (x) = ❝ã Ýt ♥❤✃t ✷ ♥❣❤✐Ư♠❀ ❞✮ P❤➢➡♥❣ tr×♥❤ f (x) = ❝ã Ýt ♥❤✃t ✸ ♥❣❤✐Ö♠✳ s❛✉ ➤➞② ❧➭ f, g ❧➭ ❝➳❝ ❤➭♠ sè ①➳❝ ➤Þ♥❤ ✈➭ ❧✐➟♥ tơ❝ tr➟♥ t❐♣ A ⊂ R✳ ❑❤➻♥❣ ➤Þ♥❤ ♥➭♦ ●✐➯ sư ❈➞✉ ✹✼✳ ➤ó♥❣❄ ❛✮ ❍➭♠ h(x) = min{f (x), g(x)} ❧✐➟♥ tô❝ tr➟♥ A❀ ❜✮ ❍➭♠ h(x) = min{f (x), g(x)} ❦❤➠♥❣ ❧✐➟♥ tô❝ tr➟♥ A❀ ❝✮ ❍➭♠ h(x) = min{f (x), g(x)} ❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥ A❀ ❞✮ ❍➭♠ h(x) = min{f (x), g(x)} ❦❤➠♥❣ ❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥ A✳ ❈➞✉ ✹✽✳ s❛✉ ➤➞② ❧➭ x∈A x∈A x∈A x∈A ●✐➯ sư f, g ❧➭ ❝➳❝ ❤➭♠ sè ①➳❝ ➤Þ♥❤ ✈➭ ❧✐➟♥ tô❝ tr➟♥ A = [a, b]✳ s❛✐❄ ❛✮ ❍➭♠ h(x) = max {f (x), g(x)} ❧✐➟♥ tô❝ tr➟♥ A❀ ❜✮ ❍➭♠ h(x) = max {f (x), g(x)} ➤➵t ❣✐➳ trÞ ♥❤á ♥❤✃t tr➟♥ A❀ ❝✮ ❍➭♠ h(x) = max {f (x), g(x)} ❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥ A❀ ❞✮ ❍➭♠ h(x) = max {f (x), g(x)} ❦❤➠♥❣ ❧✐➟♥ tô❝ ➤Ò✉ tr➟♥ A✳ ❈➞✉ ✹✾✳ s❛✉ ➤➞② ❧➭ x∈[a,b] x∈[a,b] x∈[a,b] x∈[a,b] ❈❤♦ ❤➭♠ sè f ①➳❝ ➤Þ♥❤ ✈➭ ❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥ (a, b)✱ a, b ∈ R✳ ❑❤➻♥❣ ị ú t ữ ❤➵♥ lim f (x) = A ✈➭ lim− f (x) = B ❀ x→a+ x→b ❜✮ ❍➭♠ ➤➵t ❣✐➳ trÞ ❧í♥ ♥❤✃t ✈➭ ♥❤á ♥❤✃t tr➟♥ ❝✮ ❍➭♠ ➤➵t ❝ù❝ trÞ tr➟♥ ❞✮ ❍➭♠ ❧✐➟♥ tơ❝ tr➟♥ ❈➞✉ ✺✵✳ ❑❤➻♥❣ ➤Þ♥❤ ♥➭♦ ❈❤♦ ♥➭♦ s❛✉ ➤➞② ❧➭ f [a, b]❀ [a, b]❀ [a, b]✳ ❧➭ ❤➭♠ ❧✐➟♥ tô❝ tr➟♥ R✱ lim f (x) = −∞✱ lim f (x) = +∞✳ ị x+ ú P trì f (x) = ❝ã ♥❣❤✐Ư♠❀ ❜✮ P❤➢➡♥❣ tr×♥❤ f (x) = ❝ã Ýt ♥❤✃t ✸ ♥❣❤✐Ư♠❀ ❝✮ P❤➢➡♥❣ tr×♥❤ f (x) = ❝ã ✷ ♥❣❤✐Ư♠❀ ❞✮ P❤➢➡♥❣ tr×♥❤ f (x) = ệ x ị trì ❈❤♦ f (x) ❧➭ ❤➭♠ ❧✐➟♥ tơ❝✱ ➤➡♥ ➤✐Ư✉ ♥❣➷t tr➟♥ [a, b] ✈➭ f (a).f (b) < 0✳ ❚×♠ ❦❤➻♥❣ s❛✐ tr♦♥❣ ❝➳❝ ❦❤➻♥❣ ➤Þ♥❤ s❛✉ ➤➞②❄ ❛✮ P❤➢➡♥❣ tr×♥❤ f (x) = ❝ã ➤ó♥❣ ✶ ♥❣❤✐Ư♠ t❤ù❝ tr➟♥ [a, b]❀ ❜✮ P❤➢➡♥❣ tr×♥❤ f (x) = ✈➠ ♥❣❤✐Ư♠ tr➟♥ [a, b]❀ ❝✮ ●✐➳ trÞ ❧í♥ ♥❤✃t ✈➭ ❜Ð ♥❤✃t ❝ñ❛ ❞✮ ◆Õ✉ tå♥ t➵✐ f (x) tr➟♥ [a, b] ❧➭ tå♥ t➵✐ ✈➭ tr➳✐ ❞✃✉❀ c ∈ (a, b) s❛♦ ❝❤♦ f (c) = 0✱ t❤× x = c ❧➭ ♥❣❤✐Ư♠ ❞✉② ♥❤✃t ❝đ❛ ♣❤➢➡♥❣ f (x) = tr➟♥ ➤♦➵♥ [a, b]✳ ❈➞✉ ✺✷✳ f ❈❤♦ ❧➭ ❤➭♠ ①➳❝ ➤Þ♥❤✱ ➤➡♥ ➤✐Ư✉ tr➟♥ [a, b] ọ trị ữ f (a) f (b) ì ị ú tr ị s ➤➞②❄ ❛✮ ❍➭♠ f ❧✐➟♥ tô❝ tr➟♥ ❜✮ ❍➭♠ f ❦❤➠♥❣ ❧✐➟♥ tô❝ tr➟♥ ❝✮ ❍➭♠ f ❦❤➠♥❣ ❧✐➟♥ tô❝ ➤Ị✉ tr➟♥ ❞✮ ❍➭♠ f ❦❤➠♥❣ ➤➵t ❣✐➳ trÞ ❧í♥ ♥❤✃t tr➟♥ ❈➞✉ ✺✸✳ ❛✮ ❚×♠ ♠Ư♥❤ ➤Ị ◆Õ✉ ❤➭♠ ❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥ f [a, b]❀ ➤ó♥❣ [a, b]❀ [a, b]❀ [a, b]✳ tr♦♥❣ ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ ➤➞②❄ ①➳❝ ➤Þ♥❤ ✈➭ ❧✐➟♥ tơ❝ tr➟♥ [a, +∞) ✈➭ tå♥ t➵✐ lim f (x) = l ∈ R✱ t❤× f x→+∞ [a, +∞)❀ ❜✮ ◆Õ✉ ❤➭♠ f ①➳❝ ➤Þ♥❤ ✈➭ ❧✐➟♥ tơ❝ tr➟♥ [a, +∞)✱ t❤× f ❝✮ ◆Õ✉ ❤➭♠ f ị tụ tr [a, +) tì lim f (x) = l ∈ R❀ ❞✮ ◆Õ✉ ❤➭♠ f ①➳❝ ➤Þ♥❤ ✈➭ ➤➡♥ ➤✐Ư✉ t➝♥❣ tr➟♥ ✶✶ ❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥ [a, +∞)❀ x→+∞ [a, +∞)✱ t❤× lim f (x) = +∞✳ x→+∞