Đồ thị hàm số \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg2 % da9maalaaabaWaaOaaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaOGa % eyOeI0IaaGinaaWcbeaaaOqaaiaadIhadaahaaWcbeqaaiaaikdaaa % GccqGHsislcaaI1aGaamiEaiabgUcaRiaaiAdaaaaaaa!4202! y = \frac{{\sqrt {{x^2} - 4} }}{{{x^2} - 5x + 6}}\) có tất cả bao nhiêu đường tiệm cận đứng và tiệm cận ngang ?

Ta có:\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci % GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcqGHRaWkcqGHEisP % aeqaaOWaaSaaaeaadaGcaaqaaiaadIhadaahaaWcbeqaaiaaikdaaa % GccqGHsislcaaI0aaaleqaaaGcbaGaamiEamaaCaaaleqabaGaaGOm % aaaakiabgkHiTiaaiwdacaWG4bGaey4kaSIaaG4naaaaaaa!484F! \mathop {\lim }\limits_{x \to + \infty } \frac{{\sqrt {{x^2} - 4} }}{{{x^2} - 5x + 7}}\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaC % beaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcqGHRaWk % cqGHEisPaeqaaOWaaSaaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaO % WaaOaaaeaadaWcaaqaaiaaigdaaeaacaWG4bWaaWbaaSqabeaacaaI % YaaaaaaakiabgkHiTmaalaaabaGaaGinaaqaaiaadIhadaahaaWcbe % qaaiaaisdaaaaaaaqabaaakeaacaWG4bWaaWbaaSqabeaacaaIYaaa % aOWaaeWaaeaacaaIXaGaeyOeI0YaaSaaaeaacaaI1aaabaGaamiEaa % aacqGHRaWkdaWcaaqaaiaaiAdaaeaacaWG4bWaaWbaaSqabeaacaaI % YaaaaaaaaOGaayjkaiaawMcaaaaaaaa!5250! = \mathop {\lim }\limits_{x \to + \infty } \frac{{{x^2}\sqrt {\frac{1}{{{x^2}}} - \frac{4}{{{x^4}}}} }}{{{x^2}\left( {1 - \frac{5}{x} + \frac{6}{{{x^2}}}} \right)}}\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaC % beaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcqGHRaWk % cqGHEisPaeqaaOWaaSaaaeaadaGcaaqaamaalaaabaGaaGymaaqaai % aadIhadaahaaWcbeqaaiaaikdaaaaaaOGaeyOeI0YaaSaaaeaacaaI % 0aaabaGaamiEamaaCaaaleqabaGaaGinaaaaaaaabeaaaOqaaiaaig % dacqGHsisldaWcaaqaaiaaiwdaaeaacaWG4baaaiabgUcaRmaalaaa % baGaaGOnaaqaaiaadIhadaahaaWcbeqaaiaaikdaaaaaaaaaaaa!4CDD! = \mathop {\lim }\limits_{x \to + \infty } \frac{{\sqrt {\frac{1}{{{x^2}}} - \frac{4}{{{x^4}}}} }}{{1 - \frac{5}{x} + \frac{6}{{{x^2}}}}} = 0\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci % GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcqGHsislcqGHEisP % aeqaaOWaaSaaaeaadaGcaaqaaiaadIhadaahaaWcbeqaaiaaikdaaa % GccqGHsislcaaI0aaaleqaaaGcbaGaamiEamaaCaaaleqabaGaaGOm % aaaakiabgkHiTiaaiwdacaWG4bGaey4kaSIaaG4naaaaaaa!485A! \mathop {\lim }\limits_{x \to - \infty } \frac{{\sqrt {{x^2} - 4} }}{{{x^2} - 5x + 7}}\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaC % beaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcqGHsisl % cqGHEisPaeqaaOWaaSaaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaO % WaaOaaaeaadaWcaaqaaiaaigdaaeaacaWG4bWaaWbaaSqabeaacaaI % YaaaaaaakiabgkHiTmaalaaabaGaaGinaaqaaiaadIhadaahaaWcbe % qaaiaaisdaaaaaaaqabaaakeaacaWG4bWaaWbaaSqabeaacaaIYaaa % aOWaaeWaaeaacaaIXaGaeyOeI0YaaSaaaeaacaaI1aaabaGaamiEaa % aacqGHRaWkdaWcaaqaaiaaiAdaaeaacaWG4bWaaWbaaSqabeaacaaI % YaaaaaaaaOGaayjkaiaawMcaaaaaaaa!525B! = \mathop {\lim }\limits_{x \to - \infty } \frac{{{x^2}\sqrt {\frac{1}{{{x^2}}} - \frac{4}{{{x^4}}}} }}{{{x^2}\left( {1 - \frac{5}{x} + \frac{6}{{{x^2}}}} \right)}}\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaC % beaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcqGHsisl % cqGHEisPaeqaaOWaaSaaaeaadaGcaaqaamaalaaabaGaaGymaaqaai % aadIhadaahaaWcbeqaaiaaikdaaaaaaOGaeyOeI0YaaSaaaeaacaaI % 0aaabaGaamiEamaaCaaaleqabaGaaGinaaaaaaaabeaaaOqaaiaaig % dacqGHsisldaWcaaqaaiaaiwdaaeaacaWG4baaaiabgUcaRmaalaaa % baGaaGOnaaqaaiaadIhadaahaaWcbeqaaiaaikdaaaaaaaaaaaa!4CE8! = \mathop {\lim }\limits_{x \to - \infty } \frac{{\sqrt {\frac{1}{{{x^2}}} - \frac{4}{{{x^4}}}} }}{{1 - \frac{5}{x} + \frac{6}{{{x^2}}}}}= 0\)

.Nên đồ thị hàm số có đường tiệm cận ngang là y = 0 .

Xét .\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa % aaleqabaGaaGOmaaaakiabgkHiTiaaiwdacaWG4bGaey4kaSIaaGOn % aiabg2da9iaaicdaaaa!3DEE! {x^2} - 5x + 6 = 0\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HS9aam % qaaqaabeqaaiaadIhacqGH9aqpcaaIYaaabaGaamiEaiabg2da9iaa % iodaaaGaay5waaaaaa!3ED0! \Leftrightarrow \left[ \begin{array}{l} x = 2\\ x = 3 \end{array} \right.\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci % GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIYaWaaWbaaWqa % beaacqGHRaWkaaaaleqaaOWaaSaaaeaadaGcaaqaaiaadIhadaahaa % WcbeqaaiaaikdaaaGccqGHsislcaaI0aaaleqaaaGcbaGaamiEamaa % CaaaleqabaGaaGOmaaaakiabgkHiTiaaiwdacaWG4bGaey4kaSIaaG % Onaaaaaaa!47D2! \mathop {\lim }\limits_{x \to {2^ + }} \frac{{\sqrt {{x^2} - 4} }}{{{x^2} - 5x + 6}}\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaC % beaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIYaWa % aWbaaWqabeaacqGHRaWkaaaaleqaaOWaaSaaaeaadaGcaaqaamaabm % aabaGaamiEaiabgkHiTiaaikdaaiaawIcacaGLPaaadaqadaqaaiaa % dIhacqGHRaWkcaaIYaaacaGLOaGaayzkaaaaleqaaaGcbaWaaeWaae % aacaWG4bGaeyOeI0IaaGOmaaGaayjkaiaawMcaamaabmaabaGaamiE % aiabgkHiTiaaiodaaiaawIcacaGLPaaaaaaaaa!4FB4! = \mathop {\lim }\limits_{x \to {2^ + }} \frac{{\sqrt {\left( {x - 2} \right)\left( {x + 2} \right)} }}{{\left( {x - 2} \right)\left( {x - 3} \right)}}\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaC % beaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIYaWa % aWbaaWqabeaacqGHRaWkaaaaleqaaOWaaSaaaeaadaGcaaqaaiaadI % hacqGHRaWkcaaIYaaaleqaaaGcbaWaaOaaaeaacaWG4bGaeyOeI0Ia % aGOmaaWcbeaakmaabmaabaGaamiEaiabgkHiTiaaiodaaiaawIcaca % GLPaaaaaaaaa!4898! = \mathop {\lim }\limits_{x \to {2^ + }} \frac{{\sqrt {x + 2} }}{{\sqrt {x - 2} \left( {x - 3} \right)}} = -\infty\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci % GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIYaWaaWbaaWqa % beaacqGHsislaaaaleqaaOWaaSaaaeaadaGcaaqaaiaadIhadaahaa % WcbeqaaiaaikdaaaGccqGHsislcaaI0aaaleqaaaGcbaGaamiEamaa % CaaaleqabaGaaGOmaaaakiabgkHiTiaaiwdacaWG4bGaey4kaSIaaG % Onaaaaaaa!47DD! \mathop {\lim }\limits_{x \to {2^ - }} \frac{{\sqrt {{x^2} - 4} }}{{{x^2} - 5x + 6}}\) không tồn tại.

Nên đồ thị hàm số có đường tiệm cận đứng là x = 2.

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci % GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIZaWaaWbaaWqa % beaacqGHRaWkaaaaleqaaOWaaSaaaeaadaGcaaqaaiaadIhadaahaa % WcbeqaaiaaikdaaaGccqGHsislcaaI0aaaleqaaaGcbaGaamiEamaa % CaaaleqabaGaaGOmaaaakiabgkHiTiaaiwdacaWG4bGaey4kaSIaaG % Onaaaaaaa!47D3! \mathop {\lim }\limits_{x \to {3^ + }} \frac{{\sqrt {{x^2} - 4} }}{{{x^2} - 5x + 6}}\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaC % beaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIZaWa % aWbaaWqabeaacqGHRaWkaaaaleqaaOWaaSaaaeaadaGcaaqaaiaadI % hadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaI0aaaleqaaaGcbaWa % aeWaaeaacaWG4bGaeyOeI0IaaGOmaaGaayjkaiaawMcaamaabmaaba % GaamiEaiabgkHiTiaaiodaaiaawIcacaGLPaaaaaaaaa!4AFD! = \mathop {\lim }\limits_{x \to {3^ + }} \frac{{\sqrt {{x^2} - 4} }}{{\left( {x - 2} \right)\left( {x - 3} \right)}} = +\infty\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci % GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIZaWaaWbaaWqa % beaacqGHsislaaaaleqaaOWaaSaaaeaadaGcaaqaaiaadIhadaahaa % WcbeqaaiaaikdaaaGccqGHsislcaaI0aaaleqaaaGcbaGaamiEamaa % CaaaleqabaGaaGOmaaaakiabgkHiTiaaiwdacaWG4bGaey4kaSIaaG % Onaaaaaaa!47DE! \mathop {\lim }\limits_{x \to {3^ - }} \frac{{\sqrt {{x^2} - 4} }}{{{x^2} - 5x + 6}}\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaC % beaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIZaWa % aWbaaWqabeaacqGHsislaaaaleqaaOWaaSaaaeaadaGcaaqaaiaadI % hadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaI0aaaleqaaaGcbaWa % aeWaaeaacaWG4bGaeyOeI0IaaGOmaaGaayjkaiaawMcaamaabmaaba % GaamiEaiabgkHiTiaaiodaaiaawIcacaGLPaaaaaaaaa!4B08! = \mathop {\lim }\limits_{x \to {3^ - }} \frac{{\sqrt {{x^2} - 4} }}{{\left( {x - 2} \right)\left( {x - 3} \right)}} = -\infty\)

Nên đồ thị hàm số có đường tiệm cận đứng là x = 3.

Vậy đồ thị hàm số có 3 đường tiệm cận đứng và tiệm cận ngang.