{"id":568,"date":"2025-01-27T15:53:13","date_gmt":"2025-01-27T10:08:13","guid":{"rendered":"https:\/\/pokharelsugam.com.np\/bioinformatics\/?p=568"},"modified":"2025-01-28T16:28:56","modified_gmt":"2025-01-28T10:43:56","slug":"limits-continuity","status":"publish","type":"post","link":"https:\/\/pokharelsugam.com.np\/bioinformatics\/limits-continuity\/","title":{"rendered":"Limits &amp; Continuity"},"content":{"rendered":"\n<script type=\"text\/javascript\" async \n  src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\">\n<\/script>\n\n\n\n<p class=\"has-medium-font-size\">Limit of a Constant:<br>\\( \\lim \\limits_{x \\to a} c = c \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">\\( \\lim \\limits_{x \\to a} x = a \\)<br>\\(\\lim \\limits_{x \\to a} \\frac{x^n-a^n}{x-a} =na^{n-1} \\)<br>\\(\\lim \\limits_{x \\to a} \\frac{x^m-a^m}{x^n-a^n} =\\frac{m^{m-n}}{n} \\)<br>\\(\\lim \\limits_{x \\to 0} \\frac{a^x-1}{x} = \\ln a \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Exponential Functions:<br>\\( \\lim \\limits_{x \\to a} e^x = e^a \\)<br>\\( \\lim \\limits_{x \\to 0} \\frac{e^x-1}{x} = 1  \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Logarithmic Functions:<br>\\( \\lim \\limits_{x \\to a} \\ln(x) = \\ln(a)  \\)<br>\\( \\lim \\limits_{x \\to 0} \\frac{\\ln(1+x)}{x} = 1  \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Trigonometric Functions:<br>\\( \\lim \\limits_{x \\to c} \\sin(x) = \\sin(c) \\)<br>\\( \\lim \\limits_{x \\to c} \\cos(x) = \\cos(c) \\)<br>\\( \\lim \\limits_{x \\to c} \\sec(x) = \\sec(c) \\)<br>\\( \\lim \\limits_{\\theta \\to 0} \\frac{\\sin(\\theta)}{\\theta} = 1 \\)<br>\\( \\lim \\limits_{\\theta \\to 0} \\sin(\\theta) = 0 \\)<br>\\( \\lim \\limits_{\\theta \\to 0} \\cos(\\theta) = 1 \\)<br>\\( \\lim \\limits_{x \\to 0} \\frac{1 &#8211; \\cos(x)}{x} = 0 \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Infinity Rules:<br>\\( \\lim \\limits_{x \\to \\infty} \\frac{1}{x} = 0 \\)<br>\\( \\lim \\limits_{x \\to \\infty} \\frac{1}{x^2} = 0 \\)<br>\\( \\lim \\limits_{x \\to \\infty} e^{-x} = 0 \\)<br>\\( \\lim \\limits_{x \\to \\infty} \\ln(x) = \\infty \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Special Limits:<br>\\( \\lim \\limits_{x \\to \\infty} \\left( 1 + \\frac{1}{x} \\right)^x = e \\)<br>\\( \\lim \\limits_{x \\to 0^+} x^x = 1 \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Sum\/Diff Rule:<br>\\( \\lim \\limits_{x \\to a} \\left[ f(x) \\pm g(x) \\right] = \\lim \\limits_{x \\to a} f(x) \\pm \\lim \\limits_{x \\to a} g(x)\\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Product Rule:<br>\\( \\lim \\limits_{x \\to a} \\left[ f(x) \\cdot g(x) \\right] = \\left( \\lim \\limits_{x \\to a} f(x) \\right) \\cdot \\left( \\lim \\limits_{x \\to a} g(x) \\right) \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Quotient Rule:<br>\\( \\lim \\limits_{x \\to a} \\left[ \\frac{f(x)}{g(x)} \\right] = \\frac{\\lim \\limits_{x \\to a} f(x)}{\\lim \\limits_{x \\to a} g(x)}  \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Power Rule:<br>\\( \\lim \\limits_{x \\to a} \\left[ f(x) \\right]^n = \\left( \\lim \\limits_{x \\to a} f(x) \\right)^n \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Root Rule:<br>\\( \\lim \\limits_{x \\to a} \\sqrt[n]{f(x)} = \\sqrt[n]{\\lim \\limits_{x \\to a} f(x)} \\quad \\text{if } f(x) \\geq 0 \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">L&#8217;H\u00f4pital&#8217;s Rule:<br>If \\( \\lim \\limits_{x \\to a} \\frac{f(x)}{g(x)} = \\frac{0}{0} \\text{ or } \\frac{\\pm \\infty}{\\pm \\infty} \\), then<br>\\( \\lim \\limits_{x \\to a} \\frac{f(x)}{g(x)} = \\lim \\limits_{x \\to a} \\frac{f'(x)}{g'(x)} \\),<br>provided the derivatives exist and the limit on the right exists.<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Continuity and Limits:<br>If ( f(x) ) is continuous at ( x = a ), then \\( \\lim \\limits_{x \\to a} f(x) = f(a) \\).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Limit of a Constant:\\( 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[&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":null,"ast-site-content-layout":"default","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":null,"ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":null,"ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":null,"stick-header-meta":null,"header-above-stick-meta":null,"header-main-stick-meta":null,"header-below-stick-meta":null,"astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center 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