{"id":4057,"date":"2024-01-30T12:25:28","date_gmt":"2024-01-30T06:25:28","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=4057"},"modified":"2024-11-05T13:39:26","modified_gmt":"2024-11-05T07:39:26","slug":"limit-formula","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/limit-formula\/","title":{"rendered":"\u09b2\u09bf\u09ae\u09bf\u099f\u09c7\u09b0 \u09b8\u09c2\u09a4\u09cd\u09b0 (Limit formula)\u00a0"},"content":{"rendered":"<h2><span style=\"color: #339966;\"><b>\u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a3 \u09b8\u09c2\u09a4\u09cd\u09b0: \u09b2\u09bf\u09ae\u09bf\u099f <\/b><b>(Limit formula)\u00a0<\/b><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u09a8 \u09a4\u09c7\u09a8 \u09aa\u09cd\u09b0\u0995\u09be\u09b0\u09c7\u09a8 ( \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u0989\u09aa\u09be\u09df ) \u0985\u09b8\u0982 \u099c\u09cd\u099e\u09be\u09df\u09bf\u09a4 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09df \u09a6\u09c2\u09b0 \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7<\/span><span style=\"font-weight: 400;\">\u0964<\/span><\/p>\n<h3><span style=\"color: #800080;\"><b>\u09aa\u09cd\u09b0\u09df\u09cb\u099c\u09a8\u09c0\u09df \u0995\u09bf\u099b\u09c1 \u09a7\u09be\u09b0\u09be <\/b><b>(Some necessary sections)<\/b><b>\u00a0<\/b><\/span><\/h3>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">(1+x)^{n} <\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">1+n x+\\frac{n(n-1)}{2 !} \\mathrm{x}^{2}+\\frac{\\mathrm{n}(\\mathrm{n}-1)(n-2)}{3 !} x^{3}+------- <\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">e^{x} <\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">1+x+\\frac{x^{2}}{2 !}+\\frac{x^{3}}{3 !}+\\frac{x^{4}}{4 !}+ <\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\ln (1+x) <\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">x-\\frac{x^{2}}{2}+\\frac{x^{3}}{3}-\\frac{x^{4}}{4}+------ <\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\ln (1-x) <\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">-x-\\frac{x^{2}}{2}-\\frac{x^{3}}{3}-\\frac{x^{4}}{4}------- <\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">e^{-x} <\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">1-x+\\frac{x^{2}}{2 !}-\\frac{x^{3}}{3 !}+\\frac{x^{4}}{4 !}-\\text { - - - - - - - } <\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">a^{x} <\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">1+x \\ln a+\\frac{x^{2}}{2 !}(\\ln a)^{2}+\\frac{x^{3}}{3 !}(\\ln a)^{3}+---------- <\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">a^{-x} <\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">1-x \\ln a+\\frac{x^{2}}{2 !}(\\ln a)^{2}-\\frac{x^{3}}{3 !}(\\ln a)^{3}+------- <\/span><\/li>\n<\/ul>\n<h3><span style=\"color: #800080;\"><b>\u09ac\u09bf\u09b6\u09c7\u09b7 \u0995\u09bf\u099b\u09c1 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09b8\u09c0\u09ae\u09be \u09a8\u09bf\u09b0\u09cd\u09a3\u09df <\/b><\/span><b><span style=\"color: #800080;\">(Determining the<\/span> <span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/www.youtube.com\/watch?v=Rkj9BuBoCjc&amp;list=PL0dr4HGr8HPiPsV8-EEQp7jXqdnKyjhFg\" target=\"_blank\" rel=\"noopener\">limits<\/a><\/span><span style=\"color: #800080;\"> of certain functions) :<\/span><\/b><\/h3>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\text { 1. } \\frac{\\operatorname{tin}_{i i}}{x \\rightarrow 0} \\frac{e^{x}-1}{x}=1<\/span>\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a8\u0983\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/b><span class=\"katex-eq\" data-katex-display=\"false\">e^{x}<\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">1+x+\\frac{x^{2}}{2 !}+\\frac{x^{3}}{3 !}+------- <\/span><\/p>\n<p><span class=\"katex-eq\" data-katex-display=\"false\">\\therefore \\frac{\\lim }{x \\rightarrow 0} \\frac{e^{x}-1}{x} <\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} \\frac{1}{x}\\left\\{\\left(1+x+\\frac{x^{2}}{2 !}+\\frac{x^{3}}{3 !}+----\\right)-1\\right\\} <\/span><\/p>\n<p><span class=\"katex-eq\" data-katex-display=\"false\">=\\frac{\\lim }{x \\rightarrow 0} \\frac{1}{x}\\left\\{x+\\frac{x^{2}}{2 !}+\\frac{x^{3}}{2 !}+---\\right\\}<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">=\\frac{\\lim }{x \\rightarrow 0} \\frac{1}{x} \\cdot x\\left\\{1+\\frac{x^{2}}{2 !}+\\frac{x^{3}}{3 !}+---\\right\\}<\/span> <\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">=1<\/span><\/span><span style=\"font-weight: 400;\"> \u00a0<\/span><b>[Proved]<\/b><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text { 2. } \\frac{\\lim }{x \\rightarrow \\mathbb{0}} \\frac{\\sin x}{x}=1<\/span><\/span><\/p>\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a8\u0983\u00a0<\/b><\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-4058 size-full\" src=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/12\/92.png\" alt=\"\u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a3 \u09b8\u09c2\u09a4\u09cd\u09b0\" width=\"864\" height=\"381\" srcset=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/12\/92.png 864w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/12\/92-300x132.png 300w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/12\/92-768x339.png 768w\" sizes=\"(max-width: 864px) 100vw, 864px\" \/><br \/>\n<span style=\"font-weight: 400;\">\u098f\u0996\u09be\u09a8\u09c7-<\/span><\/p>\n<p><span style=\"font-weight: 400;\">APB <\/span><span style=\"font-weight: 400;\">\u099a\u09be\u09aa<\/span><\/p>\n<p><span style=\"font-weight: 400;\">OP <\/span><span style=\"font-weight: 400;\">\u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7<\/span><\/p>\n<p><span style=\"font-weight: 400;\">A <\/span><span style=\"font-weight: 400;\">\u0993 <\/span><span style=\"font-weight: 400;\">B <\/span><span style=\"font-weight: 400;\">\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7<\/span> <span style=\"font-weight: 400;\">\u0985\u0999\u09cd\u0995\u09bf\u09a4<\/span><span style=\"font-weight: 400;\">\u00a0 <\/span><span style=\"font-weight: 400;\">\u09b8\u09cd\u09aa\u09b0\u09cd\u09b6\u0995<\/span><span style=\"font-weight: 400;\"> OP <\/span><span style=\"font-weight: 400;\">\u098f\u09b0<\/span> <span style=\"font-weight: 400;\">\u09ac\u09b0\u09cd\u09a7\u09bf\u09a4\u09be\u0982\u09b6\u0995\u09c7<\/span><span style=\"font-weight: 400;\"> c <\/span><span style=\"font-weight: 400;\">\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7<\/span> <span style=\"font-weight: 400;\">\u09aa\u09b0\u09b8\u09cd\u09aa\u09b0\u0995\u09c7<\/span> <span style=\"font-weight: 400;\">\u099b\u09c7\u09a6<\/span> <span style=\"font-weight: 400;\">\u0995\u09b0\u09c7<\/span><span style=\"font-weight: 400;\">\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">&lt;P0A=<span class=\"katex-eq\" data-katex-display=\"false\">x <\/span> <\/span><span style=\"font-weight: 400;\">\u09b0\u09c7\u09a1\u09bf\u09df\u09be\u09a8<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u0996\u09a8-<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u099c\u09cd\u09af\u09be AB &lt; \u099a\u09be\u09aa APB\u00a0 &lt; AC + BC<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<\/span><span style=\"font-weight: 400;\">2AD &lt; 2 \u099a\u09be\u09aa AP &lt; 2AC<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<\/span><span style=\"font-weight: 400;\">AD &lt; \u099a\u09be\u09aa AP &lt;AC<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{A D}{O A} <\/span>&lt;<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\text {\u099a\u09be\u09aa } A P}{O A} <\/span>&lt;<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{A C}{O A} <\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\sin x <\/span>&lt;<span class=\"katex-eq\" data-katex-display=\"false\">x <\/span>&lt;<span class=\"katex-eq\" data-katex-display=\"false\">tan x--------1 <\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<\/span>1&lt;<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{x}{\\sin x} <\/span>&lt;<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{1}{\\cos x} <\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\cos x <\/span>&lt;<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\sin x}{x} <\/span>&lt;1<\/p>\n<p><span class=\"katex-eq\" data-katex-display=\"false\">x \\rightarrow 0 <\/span> \u09b9\u09b2\u09c7,<span class=\"katex-eq\" data-katex-display=\"false\">\\cos x \\rightarrow 1 <\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} <\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\cos x <\/span>&lt;<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} <\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\sin x}{x} <\/span>&lt;1<\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<\/span>1&lt;<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} <\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\sin x}{x} <\/span>&lt;1<\/p>\n<p>\u2234<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} <\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\sin x}{x} <\/span>=1<\/p>\n<p><span style=\"font-weight: 400;\">[<\/span><b>Proved<\/b><span style=\"font-weight: 400;\">]<\/span><\/p>\n<p>&nbsp;<\/p>\n<p>3.<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\ln _{\\text {iii }}}{x \\rightarrow 0} \\frac{\\operatorname{tar} x}{x}=1<\/span><\/p>\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a8\u0983\u00a0<\/b><\/p>\n<p>(1) \u09b9\u09a4\u09c7 \u09aa\u09be\u0987 &#8211;<\/p>\n<p><span class=\"katex-eq\" data-katex-display=\"false\">\\sin x <\/span>&lt; <span class=\"katex-eq\" data-katex-display=\"false\">x <\/span> &lt; <span class=\"katex-eq\" data-katex-display=\"false\">\\tan x <\/span><\/p>\n<p>\u27f9<span class=\"katex-eq\" data-katex-display=\"false\">\\cos x <\/span>&lt; <span class=\"katex-eq\" data-katex-display=\"false\">\\frac{x}{\\tan x} <\/span> &lt; 1<\/p>\n<p>\u27f9<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} <\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\cos x <\/span>&lt;<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} <\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{x}{\\tan x} <\/span>&lt;1<\/p>\n<p>\u27f91&lt;<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} <\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{x}{\\tan x} <\/span>&lt;1<\/p>\n<p>\u2234<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} <\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{x}{\\tan x} <\/span>=1<\/p>\n<p><strong>[Proved]<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>4.<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} <\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\ln (1+x)}{x} <\/span>=1<\/p>\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a8\u0983\u00a0<\/b><\/p>\n<p><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} <\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\ln (1+x)}{x} <\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} \\frac{1}{x} <\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\left(x-\\frac{x^{2}}{2}+\\frac{x^{3}}{3}+\\frac{x^{4}}{4}+----\\right) <\/span><\/p>\n<p>=<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} <\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{1}{x} <\/span>.<span class=\"katex-eq\" data-katex-display=\"false\">x <\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\left(1-\\frac{x^{2}}{2}+\\frac{x^{3}}{3}-\\frac{x^{4}}{4}+-----\\right) <\/span><\/p>\n<p>=1<\/p>\n<p><strong>[Proved]<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>5.<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} <\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{{x^{n}-a^{n}}}{x-a} <\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">n a^{n}-1 <\/span><\/p>\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a8\u0983\u00a0<\/b><\/p>\n<p>\u09a7\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\">x=a+h ; h \\rightarrow 0<\/span> \u09af\u0996\u09a8 <span class=\"katex-eq\" data-katex-display=\"false\">x \\rightarrow a<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow a} \\frac{x^{n}-a^{n}}{x-a}=\\frac{\\lim }{h \\rightarrow 0} \\frac{(a+h)^{\\mathrm{n}}-a^{\\mathrm{n}}}{a+h-a}<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{l}\n\n=\\frac{\\lim }{h \\rightarrow 0} \\frac{(\\mathrm{a}+\\mathrm{h})^{\\mathrm{n}}-\\mathrm{a}^{\\mathrm{n}}}{\\mathrm{h}} \\\\\n\n=\\frac{\\lim }{h \\rightarrow 0} \\frac{1}{h}\\left\\{a^{n}\\left(1+\\frac{h}{a}\\right)^{n}-a^{n}\\right\\} \\\\\n\n=\\frac{\\lim }{h \\rightarrow 0} \\frac{a^{n}}{h}\\left\\{\\left(1+\\frac{h}{a}\\right)^{n}-1\\right\\} \\\\\n\n=\\frac{\\lim }{h \\rightarrow 0} \\frac{a^{n}}{h}\\left\\{1+\\frac{n h}{a}+\\frac{n(n-1)}{2 !} \\frac{h^{2}}{a^{2}}+\\frac{n(n-1)(n-2)}{3 !} \\frac{h^{3}}{a^{3}}+-----1\\right\\} \\\\\n\n=\\frac{\\lim }{h \\rightarrow 0} \\frac{a^{n}}{h} \\cdot \\frac{n h}{a}\\left\\{1+\\frac{(n-1)}{2 !} \\frac{h}{a}+\\frac{n(n-1)(n-2)}{3 !} \\frac{h^{2}}{a^{2}}+---\\right\\} \\\\\n\n=\\frac{n a^{n}}{a}=n a^{n-1}\n\n\\end{array}<\/span>\n<p><strong>[Proved]<\/strong><\/p>\n<h3><strong><span style=\"color: #800080;\">\u27f9 \u098f\u0995\u09a8\u099c\u09b0\u09c7 \u09b8\u09ac<\/span> <span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/academic\/10\/\">\u09b8\u09c2\u09a4\u09cd\u09b0\u0983<\/a><\/span><\/strong><\/h3>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} \\frac{\\sin x}{x}=1<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} \\frac{x}{\\sin x}=1<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} \\frac{\\tan x}{x}=1<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} \\frac{x}{\\tan x}=1<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} \\frac{e^{x}-1}{x}=1<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} \\frac{\\ln (1+x)}{x}=1<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} \\frac{(1+x)^{n}-1}{x}=n<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0} \\frac{x^{n}-a^{n}}{x-a}=n a^{n-1}<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow 0}(1+x)^{\\frac{1}{x}}=e<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\lim }{x \\rightarrow \\infty}\\left(1+\\frac{1}{x}\\right)^{x}=e<\/span><\/li>\n<\/ul>\n<div class=\"x1tlxs6b x1g8br2z x1gn5b1j x230xth x14ctfv x1okitfd x6ikm8r x10wlt62 x1mzt3pk x1y1aw1k xn6708d xwib8y2 x1ye3gou x1n2onr6 x13faqbe x1vjfegm\" role=\"none\">\n<div class=\"\">\n<div class=\"x9f619 x1n2onr6 x1ja2u2z __fb-light-mode\" role=\"none\">\n<hr \/>\n<p dir=\"auto\" role=\"none\">\n<p class=\"x6prxxf x1fc57z9 x1yc453h x126k92a xzsf02u\" dir=\"auto\" role=\"none\"><em><strong>\u098f\u0987\u099a\u098f\u09b8\u09b8\u09bf \u0993 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be\u09b0\u09cd\u09a5\u09c0\u09a6\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n<ul>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-25-online-batch-2-bangla-english-ict\/\">HSC 25 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a \u09e8.\u09e6 (\u09ac\u09be\u0982\u09b2\u09be, \u0987\u0982\u09b0\u09c7\u099c\u09bf, \u09a4\u09a5\u09cd\u09af \u0993 \u09af\u09cb\u0997\u09be\u09af\u09cb\u0997 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-26-online-batch-bangla-english-ict\/\">HSC 26 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ac\u09be\u0982\u09b2\u09be, \u0987\u0982\u09b0\u09c7\u099c\u09bf, \u09a4\u09a5\u09cd\u09af \u0993 \u09af\u09cb\u0997\u09be\u09af\u09cb\u0997 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-2025-online-batch\/\">HSC 25 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ab\u09bf\u099c\u09bf\u0995\u09cd\u09b8, \u0995\u09c7\u09ae\u09bf\u09b8\u09cd\u099f\u09cd\u09b0\u09bf, \u09ae\u09cd\u09af\u09be\u09a5, \u09ac\u09be\u09df\u09cb\u09b2\u099c\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-2026-online-batch\/\">HSC 26 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ab\u09bf\u099c\u09bf\u0995\u09cd\u09b8, \u0995\u09c7\u09ae\u09bf\u09b8\u09cd\u099f\u09cd\u09b0\u09bf, \u09ae\u09cd\u09af\u09be\u09a5, \u09ac\u09be\u09df\u09cb\u09b2\u099c\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/medical-admission-course\/\">\u09ae\u09c7\u09a1\u09bf\u0995\u09c7\u09b2 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/dhaka-university-a-unit-admission-course\/\">\u09a2\u09be\u0995\u09be \u09ad\u09be\u09b0\u09cd\u09b8\u09bf\u099f\u09bf A Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/dhaka-university-b-unit-admission-course\/\">\u09a2\u09be\u0995\u09be \u09ad\u09be\u09b0\u09cd\u09b8\u09bf\u099f\u09bf B Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/buet-ques-solve\/\">\u09ac\u09c1\u09df\u09c7\u099f \u0995\u09cb\u09b6\u09cd\u099a\u09c7\u09a8 \u09b8\u09b2\u09ad \u0995\u09cb\u09b0\u09cd\u09b8<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/gst-a-unit-admission-course\/\">\u0997\u09c1\u099a\u09cd\u099b A Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/gst-b-unit-admission-course\/\">\u0997\u09c1\u099a\u09cd\u099b B Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><em><strong>\u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u09b8\u09cd\u0995\u09bf\u09b2 \u09a1\u09c7\u09ad\u09c7\u09b2\u09aa\u09ae\u09c7\u09a8\u09cd\u099f \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\n<ul>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/study-abroad-complete-guideline\/\">\u09ac\u09bf\u09a6\u09c7\u09b6\u09c7 \u0989\u099a\u09cd\u099a\u09b6\u09bf\u0995\u09cd\u09b7\u09be: Study Abroad Complete Guideline<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/student-hacks\/\">Student Hacks<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ielts-course\/\">IELTS Course by Munzereen Shahid<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/english-grammar-course\/\">Complete English Grammar Course<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ms-bundle\/\"> Microsoft Office 3 in 1 Bundle<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ghore-boshe-freelancing\/\">\u0998\u09b0\u09c7 \u09ac\u09b8\u09c7 Freelancing<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/facebook-marketing\/\">Facebook Marketing<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/adobe-4-in-1-bundle\/\">Adobe 4 in 1 Bundle<\/a><\/span><\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><em>\u09e7<\/em><em>\u09e6 \u09ae\u09bf\u09a8\u09bf\u099f \u09b8\u09cd\u0995\u09c1\u09b2\u09c7\u09b0 \u0995\u09cd\u09b2\u09be\u09b8\u0997\u09c1\u09b2\u09cb \u0985\u09a8\u09c1\u09b8\u09b0\u09a3 \u0995\u09b0\u09a4\u09c7 \u09ad\u09bf\u099c\u09bf\u099f: <span style=\"color: #993300;\"><strong><a style=\"color: #993300;\" href=\"https:\/\/10minuteschool.com\/?ref=https%3A%2F%2Fblog.10minuteschool.com%2Fwordpress%2F&amp;post_id=78178&amp;blog_category_id=700\">www.10minuteschool.com<\/a><\/strong><\/span><\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a3 \u09b8\u09c2\u09a4\u09cd\u09b0: \u09b2\u09bf\u09ae\u09bf\u099f (Limit formula)\u00a0 \u09af\u09c7\u09a8 \u09a4\u09c7\u09a8 \u09aa\u09cd\u09b0\u0995\u09be\u09b0\u09c7\u09a8 ( \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u0989\u09aa\u09be\u09df ) \u0985\u09b8\u0982 \u099c\u09cd\u099e\u09be\u09df\u09bf\u09a4 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09df \u09a6\u09c2\u09b0 \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964 \u09aa\u09cd\u09b0\u09df\u09cb\u099c\u09a8\u09c0\u09df \u0995\u09bf\u099b\u09c1 \u09a7\u09be\u09b0\u09be (Some necessary sections)\u00a0 = = = = = = = \u09ac\u09bf\u09b6\u09c7\u09b7 \u0995\u09bf\u099b\u09c1 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09b8\u09c0\u09ae\u09be \u09a8\u09bf\u09b0\u09cd\u09a3\u09df (Determining the limits of<\/p>\n<p> <a class=\"redmore\" href=\"https:\/\/10minuteschool.com\/content\/limit-formula\/\">Read More<\/a><\/p>\n","protected":false},"author":9,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[4264,3037,3026],"tags":[2708,2570,2703,2705,2707,2706,2709],"_links":{"self":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/4057"}],"collection":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/users\/9"}],"replies":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/comments?post=4057"}],"version-history":[{"count":16,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/4057\/revisions"}],"predecessor-version":[{"id":16125,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/4057\/revisions\/16125"}],"wp:attachment":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/media?parent=4057"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/categories?post=4057"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/tags?post=4057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}