--- title: Asymptotes of Rational Functions - Explained Simply description: "Asymptotes of a rational function: vertical lines from the denominator, and the horizontal end behavior." url: https://astra-ai.co/videos/rational-function/graph-of-a-rational-function/asymptotes-of-a-rational-function/ locale: en type: video-lesson keywords: - asymptotes of a rational function - vertical asymptote - horizontal asymptote - rational graph published: 2026-10-01T10:09:50.920Z updated: 2026-10-04T22:00:27.947Z alternates: bg: https://astra-ai.co/bg/videos/ratsionalna-funktsiya/grafika-na-ratsionalna-funktsiya/asymptoti-na-ratsionalna-funktsiya/ br: https://astra-ai.co/br/videos/funcao-racional/capitulo-matematica-006/video-matematica-011/ cs: https://astra-ai.co/cs/videos/racionalni-funkce/graf-racionalni-funkce/asymptoty-racionalni-funkce/ da: https://astra-ai.co/da/videos/rational-funktion/graf-for-en-rational-funktion/asymptoter-for-en-rationel-funktion/ de: https://astra-ai.co/de/videos/rationale-funktion/graph-einer-rationalen-funktion/asymptote-einer-rationalen-funktion/ el: https://astra-ai.co/el/videos/riti-synartisi/grafiki-parastasi-ritis-synartisis/asymptotes-of-a-rational-function/ en: https://astra-ai.co/videos/rational-function/graph-of-a-rational-function/asymptotes-of-a-rational-function/ en-IN: https://astra-ai.co/en-IN/videos/rational-function/chapter-bf9d9f6a/asymptotes-of-a-rational-function/ en-PH: https://astra-ai.co/en-PH/videos/rational-function/chapter-bf9d9f6a/lesson-e5eb0a6c/ es: https://astra-ai.co/es/videos/funcion-racional/grafica-de-una-funcion-racional/asintotas-de-una-funcion-racional/ es-AR: https://astra-ai.co/es-AR/videos/funcion-racional/capitulo-2/leccion-11/ es-CL: https://astra-ai.co/es-CL/videos/funcion-racional/capitulo-2/asintotas-de-una-funcion-racional/ es-PE: https://astra-ai.co/es-PE/videos/funcion-racional/capitulo-video-bf9d9f6a/leccion-de-video-e5eb0a6c/ et: https://astra-ai.co/et/videos/ratsionaalfunktsioon/ratsionaalfunktsiooni-graafik/ratsionaalfunktsiooni-asumptoodid/ fi: https://astra-ai.co/fi/videos/rationaalifunktio/rationaalifunktion-kuvaaja/rationaalifunktion-asymptootit/ fr: https://astra-ai.co/fr/videos/fonction-rationnelle/graphique-dune-fonction-rationnelle/asymptotes-dune-fonction-rationnelle/ hr: https://astra-ai.co/hr/videos/racionalna-funkcija/graf-racionalne-funkcije/asimptote-racionalne-funkcije/ hu: https://astra-ai.co/hu/videos/racionalis-fuggveny/racionalis-fuggveny-grafikonja/racionalis-fuggveny-aszimptotai/ id: https://astra-ai.co/id/videos/fungsi-rasional/grafik-fungsi-rasional/asimtot-fungsi-rasional/ it: https://astra-ai.co/it/videos/funzione-razionale/grafico-funzione-razionale/asintoti-funzione-razionale/ ja: https://astra-ai.co/ja/videos/rational-function-ja/ja-chapter-02/011/ km: https://astra-ai.co/km/videos/rational-function/chapter-bf9d9f6a/lesson-e5eb0a6c/ ko: https://astra-ai.co/ko/videos/rational-function-ko/02/011/ lo: https://astra-ai.co/lo/videos/lo-cb97549a/lo-bf9d9f6a/e5eb0a6c/ lt: https://astra-ai.co/lt/videos/racionalioji-funkcija/racionaliosios-funkcijos-grafikas/pamoka-e5eb0a6c/ lv: https://astra-ai.co/lv/videos/racionala-funkcija/racionalas-funkcijas-grafiks/stunda-e5eb0a6c/ ms: https://astra-ai.co/ms/videos/fungsi-nisbah/bab-06-matematik-asas/pelajaran-011/ ms-BN: https://astra-ai.co/ms-BN/videos/fungsi-nisbah/chapter-bf9d9f6a/pelajaran-e5eb0a6c/ mx: https://astra-ai.co/mx/videos/funcion-racional/capitulo-2/leccion-11/ my: https://astra-ai.co/my/videos/my-cb97549a/my-bf9d9f6a/e5eb0a6c/ nl: https://astra-ai.co/nl/videos/rationele-functie/grafiek-van-een-rationale-functie/asymptoten-van-een-rationale-functie/ no: https://astra-ai.co/no/videos/rasjonale-funksjoner/grafen-til-en-rasjonal-funksjon/asymptoter-til-en-rasjonal-funksjon/ pl: https://astra-ai.co/pl/videos/funkcja-wymierna/wykres-funkcji-wymiernej/asymptoty-funkcji-wymiernej/ pt: https://astra-ai.co/pt/videos/funcao-racional/grafico-de-uma-funcao-racional/assintotas-de-uma-funcao-racional/ pt-TL: https://astra-ai.co/pt-TL/videos/funcao-racional/chapter-bf9d9f6a/licao-de-video-011/ ro: https://astra-ai.co/ro/videos/functii-rationale/graficul-unei-functii-rationale/lectie-video-e5eb0a6c/ sk: https://astra-ai.co/sk/videos/racionalna-funkcia/kapitola-002/lekcia-011/ sl: https://astra-ai.co/sl/videos/racionalna-funkcija/graf-racionalne-funkcije/asimptote-racionalne-funkcije/ sr: https://astra-ai.co/sr/videos/racionalna-funkcija/poglavlje-002/lekcija-011/ sv: https://astra-ai.co/sv/videos/rationella-funktioner/grafen-for-en-rationell-funktion/asymptoter-for-en-rationell-funktion/ th: https://astra-ai.co/th/videos/fangchan-trakka/bot-06-kraf-fangchan-trakka/botrian-011/ tr: https://astra-ai.co/tr/videos/rasyonel-fonksiyon/bolum-002/ders-011/ uk: https://astra-ai.co/uk/videos/uk-topic-11/uk-chapter-06/uk-asymptotes-of-a-rational-function/ vi: https://astra-ai.co/vi/videos/vi-topic-11/vi-chapter-06/vi-asymptotes-of-a-rational-function/ zh-HK: https://astra-ai.co/zh-HK/videos/shuxue-kecheng-011/shuxue-zhang-006/shuxue-kecheng-011/ zh-TW: https://astra-ai.co/zh-TW/videos/rational-function/chapter-bf9d9f6a/lesson-e5eb0a6c/ --- # Asymptotes of a Rational Function > Asymptotes of a rational function are lines the graph approaches but does not cross in the long run. This lesson finds vertical and horizontal asymptotes. ## DEFINITION OF A RATIONAL FUNCTION A rational function is a mathematical function that can be expressed as the quotient of two polynomials. This means a rational function has the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials, and Q(x) must not be the zero polynomial (Q(x) ≠ 0). ## THE CONCEPT OF AN ASYMPTOTE An asymptote of a rational function is a straight line that describes the behavior of the function's graph as x or f(x) approaches certain values but never actually reaches them. There are three types of asymptotes: vertical, horizontal, and slant (or oblique) asymptotes. - **VERTICAL ASYMPTOTES** occur when the function approaches infinity or minus infinity as x approaches a certain value. This happens when the denominator Q(x) is equal to 0, but the numerator P(x) is not equal to 0 at the same value of x (after any common factors have been canceled). - **HORIZONTAL ASYMPTOTES** describe the behavior of the function's graph as x approaches infinity or minus infinity. A horizontal asymptote occurs when the value of the function approaches a certain constant. - **SLANT (OR OBLIQUE) ASYMPTOTES** are asymptotes that are neither vertical nor horizontal. They occur when the degree of the numerator P(x) is exactly one greater than the degree of the denominator Q(x). ## HOW TO FIND THE ASYMPTOTES OF A RATIONAL FUNCTION? - **VERTICAL ASYMPTOTES:** Solve the equation Q(x) = 0. Each solution 'c' for which P(c) ≠ 0 (after simplifying the fraction P(x)/Q(x) by canceling common factors) will be the location of a vertical asymptote, x = c. - **HORIZONTAL ASYMPTOTES:** - - If the degree of the numerator P(x) is **less than** the degree of the denominator Q(x), then y = 0 is the horizontal asymptote. - If the degrees are **equal**, the horizontal asymptote is y = (leading coefficient of P(x)) / (leading coefficient of Q(x)). - If the degree of P(x) is **greater than** the degree of Q(x), there is no horizontal asymptote. - **SLANT ASYMPTOTES:** If the degree of the numerator P(x) is **exactly one greater** than the degree of the denominator Q(x), perform polynomial long division to divide P(x) by Q(x). The quotient (which will be a linear equation of the form y = mx + b) represents the slant asymptote. The remainder term should approach zero as x approaches ±∞. ## EXAMPLE: For the function f(x) = (2x² + 3x – 5) / (x – 1), let's find the asymptotes. ### **VERTICAL ASYMPTOTE:** Set the denominator x – 1 = 0, which gives x = 1. Now, check the value of the numerator P(x) = 2x² + 3x – 5 at x = 1: P(1) = 2(1)² + 3(1) – 5 = 2 + 3 – 5 = 0. Since both the numerator and the denominator are 0 at x = 1, we should factor the numerator: 2x² + 3x – 5 = (x – 1)(2x + 5). So, f(x) = \[(x – 1)(2x + 5)] / (x – 1). For x ≠ 1, f(x) = 2x + 5. Because the (x-1) factor cancels, there is no vertical asymptote at x = 1. Instead, there is a hole in the graph at x = 1. The y-coordinate of the hole is 2(1) + 5 = 7. ### **HORIZONTAL ASYMPTOTE:** The degree of the numerator (2) is greater than the degree of the denominator (1). Therefore, there is no horizontal asymptote. ### **SLANT ASYMPTOTE:** Since the degree of the numerator is exactly one greater than the degree of the denominator, we perform polynomial long division of (2x² + 3x – 5) by (x – 1). The division yields a quotient of 2x + 5 and a remainder of 0. So, f(x) = 2x + 5, for x ≠ 1. In this particular case, because the remainder is zero, the function itself simplifies to the line y = 2x + 5 (with a hole at x=1). This line is the graph of the function, rather than a line the function approaches asymptotically. (For a typical slant asymptote, the polynomial division would result in a linear quotient mx+b and a non-zero remainder R(x), so f(x) = mx + b + R(x)/Q(x), where R(x)/Q(x) approaches 0 as x approaches ±∞. Then, y = mx + b would be the slant asymptote.)