{"id":2090,"date":"2024-02-23T19:23:32","date_gmt":"2024-02-23T19:23:32","guid":{"rendered":"https:\/\/bezwarunku.pl\/?page_id=2090"},"modified":"2024-05-09T00:18:00","modified_gmt":"2024-05-08T23:18:00","slug":"asymptoty-funkcji","status":"publish","type":"page","link":"https:\/\/bezwarunku.pl\/index.php\/asymptoty-funkcji\/","title":{"rendered":"asymptoty funkcji"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"2090\" class=\"elementor elementor-2090\" data-elementor-post-type=\"page\">\n\t\t\t\t<header class=\"elementor-element elementor-element-106dace e-con-full e-flex e-con e-parent\" data-id=\"106dace\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t<div class=\"elementor-element elementor-element-efe0d75 e-con-full e-flex e-con e-child\" data-id=\"efe0d75\" data-element_type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-4e20936 elementor-widget 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class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">zarejestruj si\u0119<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/header>\n\t\t<div class=\"elementor-element elementor-element-8af5d70 e-con-full e-flex e-con e-parent\" data-id=\"8af5d70\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-21d0580 elementor-widget elementor-widget-heading\" data-id=\"21d0580\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><a href=\"https:\/\/bezwarunku.pl\/index.php\/analiza-matematyczna\/?preview_id=200&#038;preview_nonce=87c719741f&#038;preview=true\">FUNKCJE<\/a><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b686fa3 animated-slow elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"b686fa3\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;slideInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><br><br>ASYMPTOTY FUNKCJI<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-c88129c e-flex e-con-boxed e-con e-parent\" data-id=\"c88129c\" data-element_type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t<div class=\"elementor-element elementor-element-1acfea1 e-con-full e-flex e-con e-child\" data-id=\"1acfea1\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-a4a33c4 elementor-widget elementor-widget-heading\" data-id=\"a4a33c4\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">ASYMPTOTA PIONOWA<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9425ad7 elementor-widget elementor-widget-image\" data-id=\"9425ad7\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"538\" height=\"523\" src=\"https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/pionowa.png\" class=\"attachment-large size-large wp-image-2172\" alt=\"\" srcset=\"https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/pionowa.png 538w, https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/pionowa-300x292.png 300w\" sizes=\"(max-width: 538px) 100vw, 538px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c94909e elementor-widget elementor-widget-text-editor\" data-id=\"c94909e\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Asymptoty pionowe znajdujemy obliczaj\u0105c granice w punktach, kt\u00f3re zosta\u0142y odrzucone z dziedziny funkcji. Przyk\u0142adowo funkcja:<\/p><p>\\[ f(x) = \\frac{x^2 &#8211; 5}{x &#8211; 2} \\]<\/p><p>Funkcja ma dziedzin\u0119: \\( D = \\mathbb{R} \\backslash \\{2\\} \\).<\/p><p>Aby znale\u017a\u0107 asymptot\u0119 pionow\u0105, obliczamy granice:<br \/>\\[ \\lim_{{x \\to 2^-}} f(x) \\quad \\text{oraz} \\quad \\lim_{{x \\to 2^+}} f(x) \\]<\/p><p>W co najmniej jednej z tych granic powinien wynikn\u0105\u0107 wynik \\(\\pm \\infty\\). Je\u015bli w obu granicach otrzymujemy liczby, np. \\(3\\) i \\(3\\),wtedy asymptota pionowa nie istnieje.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-7ad3135 e-con-full e-flex e-con e-child\" data-id=\"7ad3135\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-f9b409f elementor-widget elementor-widget-heading\" data-id=\"f9b409f\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">ASYMPTOTA POZIOMA<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-36b3e05 elementor-widget elementor-widget-image\" data-id=\"36b3e05\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"532\" height=\"529\" src=\"https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/pozioma.png\" class=\"attachment-large size-large wp-image-2171\" alt=\"\" srcset=\"https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/pozioma.png 532w, https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/pozioma-300x298.png 300w, https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/pozioma-150x150.png 150w, https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/pozioma-100x100.png 100w\" sizes=\"(max-width: 532px) 100vw, 532px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9a88262 elementor-widget elementor-widget-text-editor\" data-id=\"9a88262\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Asymptot\u0119 poziom\u0105 znajdujemy obliczaj\u0105c granice wed\u0142ug wzor\u00f3w:<\/p><p>\\[<br \/>\\lim_{{x \\to \\pm \\infty}} \\frac{f(x)}{x} \\quad \\text{oraz} \\quad \\lim_{{x \\to \\pm \\infty}} f(x)<br \/>\\]<\/p><p>Aby znale\u017a\u0107 asymptot\u0119 poziom\u0105, granice:<br \/>\\[<br \/>\\lim_{{x \\to \\pm \\infty}} \\frac{f(x)}{x}<br \/>\\]<br \/>musz\u0105 si\u0119 r\u00f3wna\u0107 \\(0\\).<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-97be51e e-con-full e-flex e-con e-child\" data-id=\"97be51e\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-d8fdb65 elementor-widget elementor-widget-heading\" data-id=\"d8fdb65\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">ASYMPTOTA UKO\u015aNA<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c1f59ad elementor-widget elementor-widget-image\" data-id=\"c1f59ad\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"537\" height=\"529\" src=\"https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/ukosna-1.png\" class=\"attachment-large size-large wp-image-2189\" alt=\"\" srcset=\"https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/ukosna-1.png 537w, https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/ukosna-1-300x296.png 300w, https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/ukosna-1-100x100.png 100w\" sizes=\"(max-width: 537px) 100vw, 537px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-98f87a3 elementor-widget elementor-widget-text-editor\" data-id=\"98f87a3\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Asymptot\u0119 uko\u015bn\u0105 znajdujemy obliczaj\u0105c granice wed\u0142ug wzor\u00f3w:<\/p><p>\\[<br \/>\\lim_{{x \\to \\pm \\infty}} \\frac{f(x)}{x} = A \\quad \\text{oraz} \\quad \\lim_{{x \\to \\pm \\infty}} \\left[f(x) &#8211; Ax\\right] = B<br \/>\\]<\/p><p>Obliczaj\u0105c granice z wymienionych wzor\u00f3w, mo\u017cemy wyznaczy\u0107 \\(A\\) oraz \\(B\\), dzi\u0119ki kt\u00f3rym wyznaczymy asymptot\u0119 uko\u015bn\u0105 \\(y = Ax + B\\).<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-1f436dc e-flex e-con-boxed e-con e-parent\" data-id=\"1f436dc\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t<div class=\"elementor-element elementor-element-fe31715 e-con-full e-flex e-con e-child\" data-id=\"fe31715\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-cf25985 elementor-widget elementor-widget-toggle\" data-id=\"cf25985\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pmpro\"><div class=\"pmpro_card pmpro_content_message\"><div class=\"pmpro_card_content\">nie czekaj d\u0142u\u017cej, wykup pakiet i sprawd\u017a rozwi\u0105zania.<br \/><a href=\"https:\/\/bezwarunku.pl\/index.php\/wykup-dostep\/\">Przed\u0142u\u017c pakiet!<\/a><\/div><div class=\"pmpro_card_actions pmpro_font-medium\">Already a member? <a href=\"https:\/\/bezwarunku.pl\/index.php\/membership-login\/?redirect_to=https%3A%2F%2Fbezwarunku.pl%2Findex.php%2Fasymptoty-funkcji%2F\">Log in here<\/a><\/div><\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f1012d8 elementor-widget elementor-widget-heading\" data-id=\"f1012d8\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">1. Znajd\u017a wszystkie asymptoty funkcji: $$f\\left(x\\right)=\\frac x{x+4}$$<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-f670db9 e-con-full e-flex e-con e-child\" data-id=\"f670db9\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-67aea6c elementor-widget__width-initial elementor-widget elementor-widget-toggle\" data-id=\"67aea6c\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pmpro\"><div class=\"pmpro_card pmpro_content_message\"><div class=\"pmpro_card_content\">nie czekaj d\u0142u\u017cej, wykup pakiet i sprawd\u017a rozwi\u0105zania.<br \/><a href=\"https:\/\/bezwarunku.pl\/index.php\/wykup-dostep\/\">Przed\u0142u\u017c pakiet!<\/a><\/div><div class=\"pmpro_card_actions pmpro_font-medium\">Already a member? <a href=\"https:\/\/bezwarunku.pl\/index.php\/membership-login\/?redirect_to=https%3A%2F%2Fbezwarunku.pl%2Findex.php%2Fasymptoty-funkcji%2F\">Log in here<\/a><\/div><\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c9e0eab elementor-widget elementor-widget-heading\" data-id=\"c9e0eab\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">2. Znajd\u017a wszystkie asymptoty funkcji: $$f\\left(x\\right)=\\frac{1-x^2}{x-2}$$<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-0020535 e-con-full e-flex e-con e-child\" data-id=\"0020535\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-7561037 elementor-widget__width-initial elementor-widget elementor-widget-toggle\" data-id=\"7561037\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pmpro\"><div class=\"pmpro_card pmpro_content_message\"><div class=\"pmpro_card_content\">nie czekaj d\u0142u\u017cej, wykup pakiet i sprawd\u017a rozwi\u0105zania.<br \/><a href=\"https:\/\/bezwarunku.pl\/index.php\/wykup-dostep\/\">Przed\u0142u\u017c pakiet!<\/a><\/div><div class=\"pmpro_card_actions pmpro_font-medium\">Already a member? <a href=\"https:\/\/bezwarunku.pl\/index.php\/membership-login\/?redirect_to=https%3A%2F%2Fbezwarunku.pl%2Findex.php%2Fasymptoty-funkcji%2F\">Log in here<\/a><\/div><\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-265f7aa elementor-widget elementor-widget-heading\" data-id=\"265f7aa\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">3. Znajd\u017a wszystkie asymptoty funkcji: $$f\\left(x\\right)=\\frac{2x^2}{2x-1}$$<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-c054632 e-con-full e-flex e-con e-child\" data-id=\"c054632\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-8f6fa8d elementor-widget__width-initial elementor-widget elementor-widget-toggle\" data-id=\"8f6fa8d\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pmpro\"><div class=\"pmpro_card pmpro_content_message\"><div class=\"pmpro_card_content\">nie czekaj d\u0142u\u017cej, wykup pakiet i sprawd\u017a rozwi\u0105zania.<br \/><a href=\"https:\/\/bezwarunku.pl\/index.php\/wykup-dostep\/\">Przed\u0142u\u017c pakiet!<\/a><\/div><div class=\"pmpro_card_actions pmpro_font-medium\">Already a member? <a href=\"https:\/\/bezwarunku.pl\/index.php\/membership-login\/?redirect_to=https%3A%2F%2Fbezwarunku.pl%2Findex.php%2Fasymptoty-funkcji%2F\">Log in here<\/a><\/div><\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-25403ad elementor-widget elementor-widget-heading\" data-id=\"25403ad\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">4. Znajd\u017a wszystkie asymptoty funkcji: $$f\\left(x\\right)=\\frac{x^2-3}{x^3-1}$$<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-e630129 e-con-full e-flex e-con e-child\" data-id=\"e630129\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-2f65bbd elementor-widget__width-initial elementor-widget elementor-widget-toggle\" data-id=\"2f65bbd\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pmpro\"><div class=\"pmpro_card pmpro_content_message\"><div class=\"pmpro_card_content\">nie czekaj d\u0142u\u017cej, wykup pakiet i sprawd\u017a rozwi\u0105zania.<br \/><a href=\"https:\/\/bezwarunku.pl\/index.php\/wykup-dostep\/\">Przed\u0142u\u017c pakiet!<\/a><\/div><div class=\"pmpro_card_actions pmpro_font-medium\">Already a member? <a href=\"https:\/\/bezwarunku.pl\/index.php\/membership-login\/?redirect_to=https%3A%2F%2Fbezwarunku.pl%2Findex.php%2Fasymptoty-funkcji%2F\">Log in here<\/a><\/div><\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6bc4254 elementor-widget elementor-widget-heading\" data-id=\"6bc4254\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">5. Znajd\u017a wszystkie asymptoty funkcji: $$f\\left(x\\right)=\\frac{x^2+6x}{x+2}$$<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-7888b63 e-con-full e-flex e-con e-child\" data-id=\"7888b63\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-9525735 elementor-widget__width-initial elementor-widget elementor-widget-toggle\" data-id=\"9525735\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pmpro\"><div class=\"pmpro_card pmpro_content_message\"><div class=\"pmpro_card_content\">nie czekaj d\u0142u\u017cej, wykup pakiet i sprawd\u017a rozwi\u0105zania.<br \/><a href=\"https:\/\/bezwarunku.pl\/index.php\/wykup-dostep\/\">Przed\u0142u\u017c pakiet!<\/a><\/div><div class=\"pmpro_card_actions pmpro_font-medium\">Already a member? <a href=\"https:\/\/bezwarunku.pl\/index.php\/membership-login\/?redirect_to=https%3A%2F%2Fbezwarunku.pl%2Findex.php%2Fasymptoty-funkcji%2F\">Log in here<\/a><\/div><\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-32f1bc3 elementor-widget elementor-widget-heading\" data-id=\"32f1bc3\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">6. Znajd\u017a wszystkie asymptoty funkcji: $$f\\left(x\\right)=\\frac{2x}{1-3x}$$<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-90998aa e-con-full e-flex e-con e-child\" data-id=\"90998aa\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-ade916c elementor-widget__width-initial elementor-widget elementor-widget-toggle\" data-id=\"ade916c\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pmpro\"><div class=\"pmpro_card pmpro_content_message\"><div class=\"pmpro_card_content\">nie czekaj d\u0142u\u017cej, wykup pakiet i sprawd\u017a rozwi\u0105zania.<br \/><a href=\"https:\/\/bezwarunku.pl\/index.php\/wykup-dostep\/\">Przed\u0142u\u017c pakiet!<\/a><\/div><div class=\"pmpro_card_actions pmpro_font-medium\">Already a member? <a href=\"https:\/\/bezwarunku.pl\/index.php\/membership-login\/?redirect_to=https%3A%2F%2Fbezwarunku.pl%2Findex.php%2Fasymptoty-funkcji%2F\">Log in here<\/a><\/div><\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7d9c928 elementor-widget elementor-widget-heading\" data-id=\"7d9c928\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">7. Znajd\u017a wszystkie asymptoty funkcji: $$f\\left(x\\right)=\\frac{2x^3-5x^2+x-4}{4-x^2}$$<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-b9bced9 e-con-full e-flex e-con e-child\" data-id=\"b9bced9\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-78f38ee elementor-widget__width-initial elementor-widget elementor-widget-toggle\" data-id=\"78f38ee\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pmpro\"><div class=\"pmpro_card pmpro_content_message\"><div class=\"pmpro_card_content\">nie czekaj d\u0142u\u017cej, wykup pakiet i sprawd\u017a rozwi\u0105zania.<br \/><a href=\"https:\/\/bezwarunku.pl\/index.php\/wykup-dostep\/\">Przed\u0142u\u017c pakiet!<\/a><\/div><div class=\"pmpro_card_actions pmpro_font-medium\">Already a member? <a href=\"https:\/\/bezwarunku.pl\/index.php\/membership-login\/?redirect_to=https%3A%2F%2Fbezwarunku.pl%2Findex.php%2Fasymptoty-funkcji%2F\">Log in here<\/a><\/div><\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3aa7e3f elementor-widget elementor-widget-heading\" data-id=\"3aa7e3f\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">8. Znajd\u017a wszystkie asymptoty funkcji: $$f\\left(x\\right)=\\frac{2x^2+x+1}{8x}$$<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>zarejestruj si\u0119 FUNKCJE ASYMPTOTY FUNKCJI ASYMPTOTA PIONOWA Asymptoty pionowe znajdujemy obliczaj\u0105c granice w punktach, kt\u00f3re zosta\u0142y odrzucone z dziedziny funkcji. Przyk\u0142adowo funkcja: [ f(x) = frac{x^2 &#8211; 5}{x &#8211; 2} ] Funkcja ma dziedzin\u0119: ( D = mathbb{R} backslash {2} ). Aby znale\u017a\u0107 asymptot\u0119 pionow\u0105, obliczamy granice:[ lim_{{x to 2^-}} f(x) quad text{oraz} quad lim_{{x [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"templates\/landing.php","meta":{"pmpro_default_level":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_uf_show_specific_survey":0,"_uf_disable_surveys":false,"footnotes":""},"class_list":["post-2090","page","type-page","status-publish","hentry","pmpro-has-access","entry","owp-thumbs-layout-horizontal","owp-btn-normal","owp-tabs-layout-horizontal","has-no-thumbnails","has-product-nav"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/bezwarunku.pl\/index.php\/wp-json\/wp\/v2\/pages\/2090","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/bezwarunku.pl\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/bezwarunku.pl\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/bezwarunku.pl\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/bezwarunku.pl\/index.php\/wp-json\/wp\/v2\/comments?post=2090"}],"version-history":[{"count":451,"href":"https:\/\/bezwarunku.pl\/index.php\/wp-json\/wp\/v2\/pages\/2090\/revisions"}],"predecessor-version":[{"id":4510,"href":"https:\/\/bezwarunku.pl\/index.php\/wp-json\/wp\/v2\/pages\/2090\/revisions\/4510"}],"wp:attachment":[{"href":"https:\/\/bezwarunku.pl\/index.php\/wp-json\/wp\/v2\/media?parent=2090"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}