{"id":17,"date":"2023-09-17T11:09:22","date_gmt":"2023-09-17T11:09:22","guid":{"rendered":"https:\/\/mathority.org\/id\/sifat-hukum-batas\/"},"modified":"2023-09-17T11:09:22","modified_gmt":"2023-09-17T11:09:22","slug":"sifat-hukum-batas","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/sifat-hukum-batas\/","title":{"rendered":"Properti (atau hukum) batas"},"content":{"rendered":"<p>Di sini Anda akan menemukan semua sifat (atau hukum) limit fungsi. Properti ini berfungsi untuk menyederhanakan perhitungan limit, terutama ketika menangani limit dengan operasi fungsi. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcuales-son-las-propiedades-o-leyes-de-los-limites-de-funciones\"><\/span> Apa sifat (atau hukum) limit fungsi?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Selanjutnya akan dijelaskan semua sifat-sifat limit fungsi, atau disebut juga hukum limit fungsi. Selain itu, Anda akan dapat melihat latihan yang diselesaikan untuk setiap properti limit sehingga Anda dapat memahami konsepnya sepenuhnya.<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-una-suma\"><\/span> Sifat limit suatu jumlah <span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p style=\"text-align:left\"> <strong>Limit jumlah dua fungsi<\/strong> pada suatu titik sama dengan jumlah limit masing-masing fungsi pada titik yang sama secara terpisah.<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e1975fcd0e98e2022e29be694fcdb925_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a} \\Bigl[ f(x)+g(x)\\Bigr]=\\lim_{x\\to a}f(x)+\\lim_{x\\to a}g(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"310\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Misalnya, ada dua fungsi:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1730973da231e978e4c565c939633b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2\\qquad g(x)=2x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"217\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Limit setiap fungsi di x sama dengan 1 adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-540347d33b8f0dc2d46cbc8d6f42df12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 1}x^2=1^2=1\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"121\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9e8d60234a5aaad645bcea92331ffec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 1}(2x+1)=2\\cdot1+1=3\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"209\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Oleh karena itu, limit dua fungsi yang dijumlahkan pada titik yang sama menghasilkan 4 (1+3=4).<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dee2fbc7ec3a4f0a145e79c1f8bf9ae2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 1} \\Bigl[ f(x)+g(x)\\Bigr]=\\\\[3ex]\\displaystyle =\\lim_{x\\to 1}f(x)+\\lim_{x\\to 1}g(x)=\\\\[3ex]=1+3=4\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"112\" width=\"189\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Sifat tersebut dapat dibuktikan dengan menghitung limitnya selangkah demi selangkah: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-14c858a555a3d57a24cc5c81adeda2fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 1} \\Bigl[ f(x)+g(x)\\Bigr]=\\\\[3ex]\\displaystyle =\\lim_{x\\to 1}\\Bigl[x^2+2x+1\\Bigr]=\\\\[3ex]=1^2+2\\cdot 1+1=4\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"117\" width=\"171\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-una-resta\"><\/span> Properti batas pengurangan <span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p style=\"text-align:left\"> <strong>Limit pengurangan (atau selisih) dua fungsi<\/strong> pada suatu titik sama dengan pengurangan limit masing-masing fungsi pada titik yang sama secara terpisah.<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-444c20b463064780542e57269cfd770e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a} \\Bigl[ f(x)-g(x)\\Bigr]=\\lim_{x\\to a}f(x)-\\lim_{x\\to a}g(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"310\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Menggunakan fungsi dari contoh sebelumnya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1730973da231e978e4c565c939633b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2\\qquad g(x)=2x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"217\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Limit masing-masing fungsi di titik x=3 adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f37267a3dc92ccef5a66ccfb68211919_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 3}x^2=3^2=9\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"122\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-64b82eccca7888756f96190e6374281a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 3}(2x+1)=2\\cdot3+1=7\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"209\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Maka limit kedua fungsi yang dikurangi pada x=3 adalah selisih nilai yang diperoleh pada langkah sebelumnya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-85e829be6e72a7ab7bd91d442c89558f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 3} \\Bigl[ f(x)-g(x)\\Bigr]=\\\\[3ex]\\displaystyle =\\lim_{x\\to 3}f(x)-\\lim_{x\\to 3}g(x)=\\\\[3ex]=9-7=2\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"110\" width=\"189\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Kita dapat membuktikan sifat limit ini dengan menghitung pengurangan fungsi dan kemudian menyelesaikan limitnya: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-411bbd6ad4e5d44322ec93ba06fd3088_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 3} \\Bigl[ f(x)-g(x)\\Bigr]=\\\\[3ex]\\displaystyle =\\lim_{x\\to 3}\\Bigl[x^2-(2x+1)\\Bigr]=\\\\[3ex]\\displaystyle =\\lim_{x\\to 3}\\Bigl[x^2-2x-1\\Bigr]\\\\[3ex]=3^2-2\\cdot 3-1=2\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"165\" width=\"185\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-de-limite-de-un-producto\"><\/span> Batasi properti suatu produk <span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p style=\"text-align:left\"> <strong>Limit hasil kali dua fungsi<\/strong> pada suatu titik adalah hasil kali limit masing-masing fungsi pada titik tersebut.<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d513f66b066a53d8dbd18868f1edcaa3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a} \\Bigl[ f(x)\\cdot g(x)\\Bigr]=\\lim_{x\\to a}f(x)\\cdot \\lim_{x\\to a}g(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"292\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Misalnya, jika kita memiliki dua fungsi berbeda berikut ini:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c48ccc2acf27d4fb1519d80b4c0cbe1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3\\qquad g(x)=x^2-5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"216\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Limit masing-masing fungsi pada x=2 adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eb87f9cfecaec2126043e52053704c82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 2}x^3=2^3=8\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"122\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c62f8966fd75151d6bdc8955458d1557_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 2}(x^2-5)=2^2-5=-1\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"207\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Jadi, untuk menentukan limit hasil kali kedua fungsi tersebut tidak perlu dikalikan, tetapi cukup mengalikan hasil yang diperoleh dari masing-masing limit:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0df37e873bf968f39617b8dce74edb9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 2} \\Bigl[ f(x)\\cdot g(x)\\Bigr]=\\\\[3ex]\\displaystyle =\\lim_{x\\to 2}f(x)\\cdot \\lim_{x\\to 2}g(x)=\\\\[3ex]=8\\cdot (-1)=-8\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"115\" width=\"180\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Hal ini menghemat waktu dan perhitungan karena mengalikan dua fungsi bisa jadi sulit. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-un-cociente\"><\/span> Sifat limit suatu hasil bagi <span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p style=\"text-align:left\"> <strong>Limit hasil bagi (atau pembagian) dua fungsi<\/strong> sama dengan hasil bagi limit fungsi tersebut.<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dbb038d24d534af7fc12466761b1206c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a} \\left[\\frac{f(x)}{g(x)}\\right]=\\frac{\\displaystyle\\lim_{x\\to a}f(x)}{\\displaystyle\\lim_{x\\to a}g(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"181\" style=\"vertical-align: -24px;\"><\/p>\n<\/p>\n<p> Kondisi ini terpenuhi selama limit fungsi penyebutnya tidak nol.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-615afb4e193ec9b817d1672cb66f67b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a}g(x)\\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"97\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Kita akan memecahkan contoh sifat (atau hukum) limit ini. Perhatikan fungsi f(x) dan g(x):<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-193a456033487c8b6501aa861a1b6351_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=5x-1\\qquad g(x)=3^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"217\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Pertama-tama kita hitung limit setiap fungsi pada x=0:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a25aaef5c60bc849e2cfe326093a82fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 0}(5x-1)=5\\cdot 0-1=-1\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"222\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-40be0b69270ff2804525ec6aa35a934c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 0}3^x=3^0=1\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"121\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Dengan demikian limit pembagian kedua fungsi pada x=0 dapat dengan mudah dicari:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e8558b623352036e3af2e68e72861b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 0} \\left[\\frac{f(x)}{g(x)}\\right]=\\frac{\\displaystyle\\lim_{x\\to 0}f(x)}{\\displaystyle\\lim_{x\\to 0}g(x)}=\\displaystyle\\frac{-1}{1}=-1\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"278\" style=\"vertical-align: -24px;\"><\/p>\n<\/p>\n<p> Dalam hal ini, kita dapat menerapkan properti ini untuk menyelesaikan limit karena limit g(x) bukan nol. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-una-constante\"><\/span> Sifat limit suatu konstanta <span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p style=\"text-align:left\"> <strong>Limit suatu fungsi konstanta<\/strong> selalu menghasilkan konstanta itu sendiri, terlepas dari titik di mana limit tersebut dihitung.<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-083ac97d4077fd32108f371887a6daef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a} k=k\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"74\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Properti ini sangat mudah untuk diperiksa, misalnya jika kita memiliki fungsi konstan berikut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e9e41bd168999d7634188ca08496465a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=5\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Logikanya, limit fungsi konstanta di suatu titik adalah 5: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cc495627062fe120c4fba7c649b94ea8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 0}5=5\\qquad\\qquad\\lim_{x\\to 3}5=5\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"218\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bf06e3dadceb69c684be4e04205744f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to -2}5=5\\qquad\\qquad\\lim_{x\\to 7}5=5\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"229\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-un-multiplo-constante\"><\/span> Sifat limit kelipatan konstan<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Dari sifat-sifat limit suatu hasil kali dan batas suatu konstanta, kita dapat menyimpulkan sifat-sifat berikut: <\/p>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p style=\"text-align:left\"> <strong>Limit suatu fungsi dikalikan dengan suatu konstanta<\/strong> sama dengan hasil kali konstanta tersebut dan limit fungsi tersebut.<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9e11e20d92c504cfb48a1a5c3747066f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a}\\Bigl[ k\\cdot f(x)\\Bigr]=k\\cdot\\lim_{x\\to a}f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"214\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Perhatikan bagaimana kita menyederhanakan penghitungan limit berikut menggunakan properti ini: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-422e706a1b7643d78c0d667e17ede188_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle \\lim_{x\\to 4} (2x^2-12x+10)=\\\\[3ex]\\displaystyle =\\lim_{x\\to 4}\\Bigl[2\\cdot(x^2-6x+5)\\Bigr]=\\\\[3ex]=\\displaystyle 2\\cdot\\lim_{x\\to 4}(x^2-6x+5)=\\\\[3ex]=2\\cdot (4^2-6\\cdot4+5)=\\\\[3ex]=2\\cdot (-3)=-6\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"206\" width=\"206\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-una-potencia\"><\/span> Sifat batas suatu pangkat <span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p style=\"text-align:left\"> <strong>Limit suatu fungsi yang dipangkatkan menjadi eksponen<\/strong> sama dengan menghitung limit fungsi tersebut dan kemudian menaikkan hasil limit tersebut ke eksponen tersebut.<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-02b8185a9eca6e85f4d3e69b41b09903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a}\\Bigl[f(x)^k\\Bigr]=\\left[\\lim_{x\\to a}f(x)\\right]^k\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"202\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Misalnya, limit suatu fungsi linier adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-73523d73bd77b077f64d5bf05bd0444c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 6}x=6\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"74\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Nah, limit fungsi kuadrat bisa dihitung dengan mencari limit fungsi linier lalu mengkuadratkan hasilnya: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-051ef387f34fdcc7468d761e2d618f69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 6}\\Bigl[x^2\\Bigr]=\\left[\\lim_{x\\to 6}x\\right]^2=\\bigl[6\\bigr]^2=36\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"249\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-una-funcion-exponencial\"><\/span> Sifat limit fungsi eksponensial <span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p style=\"text-align:left\"> <strong>Limit suatu fungsi eksponensial<\/strong> sama dengan konstanta fungsi yang dipangkatkan ke limit ekspresi aljabar fungsi tersebut.<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a2fc0c424758574c4b22a93c3e5dfdcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to a}\\Bigl[k^{g(x)}\\Bigr]=k^{^{\\displaystyle\\lim_{x\\to a}g(x)}}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"179\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Kami kemudian akan menghitung limit fungsi eksponensial dengan dua cara yang mungkin untuk memverifikasi properti ini: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6eb8d552dbdff46188019ca7d717e5a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 1}5^{2x}=5^{2\\cdot 1}=25\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"147\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1088a10d675c2a6dc332c23a362fce2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 1}5^{2x}=5^{^{\\displaystyle\\lim_{x\\to 1}2x}}=5^{2\\cdot 1}=25\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"232\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-una-potencia-de-funciones\"><\/span> Sifat limit suatu pangkat fungsi <span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p style=\"text-align:left\"> <strong>Limit suatu fungsi yang dipangkatkan ke fungsi lain<\/strong> adalah limit fungsi pertama yang dipangkatkan ke limit fungsi kedua.<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-11395cf067e1c7ff75ae68090bec8d16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a}\\Bigl[f(x)^{g(x)}\\Bigr]=\\left[\\lim_{x\\to a}f(x)\\right]^{\\displaystyle\\lim_{x\\to a}g(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"277\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sebagai contoh, kita akan menentukan limit berikut dengan menerapkan hukum ini: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88cf4a1a6c04abab530e67f4f0ca950d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 2}\\Bigl[(x^2-4x)^{4x-5}\\Bigr]=\\\\[3ex]\\displaystyle =\\left[\\lim_{x\\to 2}(x^2-4x)\\right]^{\\displaystyle\\lim_{x\\to 2}(4x-5)}=\\\\[3ex]=\\displaystyle (2^2-4\\cdot 2)^{4\\cdot 2-5}=\\\\[3ex]=(-4)^3=-64\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"173\" width=\"247\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-una-funcion-irracional\"><\/span> Sifat limit fungsi irasional <span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p style=\"text-align:left\"> <strong>Limit suatu akar (atau radikal)<\/strong> sama dengan akar dari limit tersebut.<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2cdf7e1a72f2441f72240f068edc413e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to a}\\sqrt[n]{f(x)}=\\sqrt[n]{\\lim_{x\\to a}f(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"197\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Untuk menggunakan properti ini, perlu diingat bahwa jika indeks akar genap, limit fungsi harus lebih besar atau sama dengan 0:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-31077c9693c2318fba7cb2f0f99d5efd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{si } n \\text{ es par} \\ \\longrightarrow \\ \\displaystyle\\lim_{x\\to a}f(x)\\ge 0\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"230\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Perhatikan bagaimana batas berikut dihitung dengan menerapkan rumus ini: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ad822bfcdd6fbb0335e1d99db71776c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 4}\\sqrt[3]{\\frac{x^2}{2}}=\\sqrt[3]{\\lim_{x\\to 4}\\frac{x^2}{2}}=\\sqrt[3]{\\frac{4^2}{2}}=\\sqrt[3]{8}=2\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"310\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-una-funcion-logaritmica\"><\/span> Sifat limit fungsi logaritma <span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p style=\"text-align:left\"> <strong>Limit suatu logaritma<\/strong> setara dengan logaritma dasar limit yang sama.<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6e34ee6ca0162d42714f0c8f3e245b14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to a}\\Bigl[\\log_k f(x)\\Bigr]=\\log_k \\left[\\lim_{x\\to a}f(x)\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"252\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Lihatlah resolusi batas berikut di mana kita menerapkan properti ini:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fc5468c73e74d00d2ffeb2293ea8145e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to -4}\\Bigl[\\log_3 (x^2-2x+3)\\Bigr]=\\\\[3ex]=\\displaystyle\\log_3 \\left[\\lim_{x\\to -4}(x^2-2x+3)\\right]=\\\\[4ex]=\\displaystyle\\log_3\\bigl[(-4)^2-2\\cdot (-4)+3\\bigr]=\\\\[3ex]=\\displaystyle\\log_3 27=3\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"177\" width=\"236\" style=\"vertical-align: 0px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Di sini Anda akan menemukan semua sifat (atau hukum) limit fungsi. Properti ini berfungsi untuk menyederhanakan perhitungan limit, terutama ketika menangani limit dengan operasi fungsi. Apa sifat (atau hukum) limit fungsi? Selanjutnya akan dijelaskan semua sifat-sifat limit fungsi, atau disebut juga hukum limit fungsi. Selain itu, Anda akan dapat melihat latihan yang diselesaikan untuk setiap &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/sifat-hukum-batas\/\"> <span class=\"screen-reader-text\">Properti (atau hukum) batas<\/span> Selengkapnya &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[43],"tags":[],"class_list":["post-17","post","type-post","status-publish","format-standard","hentry","category-batasan-fungsi"],"yoast_head":"<!-- This site is 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