{"id":23281,"date":"2021-11-08T11:53:01","date_gmt":"2021-11-08T06:23:01","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=23281"},"modified":"2024-07-07T20:50:11","modified_gmt":"2024-07-07T15:20:11","slug":"limits","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=23281","title":{"rendered":"LIMITS (Text)"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Definition}:\\ \\hspace{20cm}\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[When\\ the\\ variable\\ x\\ approaches\\ a\\ constant\\ and\\ if\\ the\\ function\\ f ( x )\\ approaches\\ a\\ constant\\  l,\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[then\\ l\\ is\\ called\\ the\\ limit\\ value\\ of\\ f ( x )\\ as\\ x\\ approaches\\ a\\ and\\ is\\ denoted\\ as\\ \\lim\\ _{x\\ \\to\\ a}\\ f(x)\\ =\\ l.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Properties}:\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1)\\ \\lim\\ _{x\\ \\to\\ a}\\  [f(x)\\ \\pm\\ g(x)]\\ =\\ lim\\ _{x\\ \\to\\ a}\\ f(x)\\ \\pm\\ lm\\ _{x\\ \\to\\  a}\\  g(x)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2)\\ \\lim\\ _{x\\ \\to\\ a}\\  [K\\ f(x)\\ ]\\ =\\ K\\ \\lim\\ _{x\\ \\to\\ a}\\ f(x)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3)\\ \\lim\\ _{x\\ \\to\\ a}\\  [f(x)\\  .\\ g(x)]\\ =\\ lim\\ _{x\\ \\to\\ a}\\ f(x)\\ .  \\ lm\\ _{x\\ \\to\\  a}\\  g(x)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4)\\ \\lim\\ _{x\\ \\to\\ a}\\ [\\frac{f(x)}{g(x)}]\\ =\\ \\frac{\\lim\\ _{x\\ \\to\\ a}\\ f(x)}{\\ lm\\ _{x\\ \\to\\  a}\\  g(x)}\\ provided\\  \\ lm\\ _{x\\ \\to\\  a}\\  g(x)\\ \\neq\\ 0.\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- billboard -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:970px;height:250px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9933277607\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 1:}\\  \\color {red} {Evaluate:}\\ Lt\\ _{x\\ \\to\\ -\\ 2}\\ (x^4\\ -\\ 3x\\ +\\ 2)(x\\ -\\ 1)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Lt\\ _{x\\ \\to\\ -\\ 2}\\ (x^4\\ -\\ 3x\\ +\\ 2)(x\\ -\\ 1)\\ =\\ ((-2)^4\\ -\\ 3(-2)\\ +\\ 2)(-\\ 2\\ -\\ 1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\ (16\\ +\\ 6\\ +\\ 2)(-\\ 3)\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\ (24)(-\\ 3)\\ \\hspace{4cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\ -\\ 72\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Lt\\ _{x\\ \\to\\ -\\ 2}\\ (x^4\\ -\\ 3x\\ +\\ 2)(x\\ -\\ 1)\\ =\\ -\\ 72}\\ \\hspace{7cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Formulae}:\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ \\lim\\ _{x\\ \\to\\ a}\\ \\frac{x^n\\ -\\ a^n}{x\\ -\\ a}\\ =\\  n\\ a^{n\\ -\\ 1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ \\lim\\ _{\u03b8\\ \\to\\ 0}\\ \\frac{Sin\\ n\u03b8}{\u03b8}\\ =\\  n\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 2:}\\  \\color {red} {Evaluate:}\\ \\lim\\ _{x\\ \\to\\ a}\\ \\frac{x^3\\ -\\ a^3}{x\\ -\\ a}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  W.\\ K.\\ T\\ \\lim\\ _{x\\ \\to\\ a}\\ \\frac{x^n\\ -\\ a^n}{x\\ -\\ a}\\ =\\  n\\ a^{n\\ -\\ 1}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\lim\\ _{x\\ \\to\\ a}\\ \\frac{x^3\\ -\\ a^3}{x\\ -\\ a}\\ =\\ 3\\ a^{3\\ -\\ 1}\\ =\\ 3\\ a^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 3.}\\ \\color {red} {Evaluate:}\\ \\lim\\ _{x\\ \\to\\ 2}\\ \\frac{x^2\\ -\\ 2^2}{x\\ -\\ 2}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ Feb\\ 2002,\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  W.\\ K.\\ T\\ \\lim\\ _{x\\ \\to\\ a}\\ \\frac{x^n\\ -\\ a^n}{x\\ -\\ a}\\ =\\  n\\ a^{n\\ -\\ 1}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\lim\\ _{x\\ \\to\\ 2}\\ \\frac{x^2\\ -\\ a^2}{x\\ -\\ 2}\\ =\\ 2\\ 2^{2\\ -\\ 1}\\ =\\ 2\\ 2^1\\ =\\ 2(2)\\ =\\ 4\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 4:}\\  \\color {red} {Evaluate:}\\ \\lim\\ _{x\\ \\to\\ 2}\\ \\frac{x^4\\ -\\ 2^4}{x\\ -\\ 2}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  W.\\ K.\\ T\\ \\lim\\ _{x\\ \\to\\ a}\\ \\frac{x^n\\ -\\ a^n}{x\\ -\\ a}\\ =\\  n\\ a^{n\\ -\\ 1}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\lim\\ _{x\\ \\to\\ 2}\\ \\frac{x^4\\ -\\ a^4}{x\\ -\\ 2}\\ =\\ 4\\ 2^{4\\ -\\ 1}\\ =\\ 4\\ 2^3\\ =\\ 4(8)\\ =\\ 32\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"853\" height=\"480\" src=\"https:\/\/www.youtube.com\/embed\/l3q59SES8Is\" title=\"Limits  - Part - I\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<script async src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\"\n     crossorigin=\"anonymous\"><\/script>\n<!-- Ad1 -->\n<ins class=\"adsbygoogle\"\n     style=\"display:block\"\n     data-ad-client=\"ca-pub-9453835310745500\"\n     data-ad-slot=\"8240817448\"\n     data-ad-format=\"auto\"\n     data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 5:}\\ \\color {red} {Evaluate:}\\ \\lim\\ _{x\\ \\to\\ 2}\\ \\frac{x^5\\ -\\ 2^5}{x^3\\ -\\ 2^3}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  W.\\ K.\\ T\\ \\lim\\ _{x\\ \\to\\ a}\\ \\frac{x^n\\ -\\ a^n}{x\\ -\\ a}\\ =\\  n\\ a^{n\\ -\\ 1}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Multiply\\ and\\ divide\\ by\\ \u2018x &#8211; 2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\lim\\ _{x\\ \\to\\ 2}\\ \\frac{\\frac{x^5\\ -\\ 2^5}{x\\ -\\ 2}}{\\frac{x^3\\ -\\ 2^3}{x\\ -\\ 2}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5\\ 2^{5\\ -\\ 1}}{3\\ 2^{3\\ -\\ 1}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5\\ 2^4}{3\\ 2^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5\\ 2^{4\\ -\\ 2}}{3}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5\\ (2^2)}{3}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5\\ (4)}{3}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{20}{3}\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"853\" height=\"480\" src=\"https:\/\/www.youtube.com\/embed\/-3YwQzuO4O8\" title=\"Limits - Part - 2\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 6:}\\  \\color {red} {Evaluate:}\\ \\lim\\ _{x\\ \\to\\ 3}\\ \\frac{x^3\\ -\\ 3^3}{x^4\\ -\\ 3^4}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  W.\\ K.\\ T\\ \\lim\\ _{x\\ \\to\\ a}\\ \\frac{x^n\\ -\\ a^n}{x\\ -\\ a}\\ =\\  n\\ a^{n\\ -\\ 1}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Multiply\\ and\\ divide\\ by\\ \u2018x &#8211; 3&#8217;\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\lim\\ _{x\\ \\to\\ 3}\\ \\frac{\\frac{x^3\\ -\\ 3^3}{x\\ -\\ 3}}{\\frac{x^4\\ -\\ 3^4}{x\\ -\\ 3}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{3\\ (3^{3\\ -\\ 1})}{4\\ (3^{4\\ -\\ 1})}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{3\\ 3^2}{4\\ 3^3}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{3}{4}\\ 3^{2\\ -\\ 3}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{3}{4}\\ 3^{-1}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{4}\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"853\" height=\"480\" src=\"https:\/\/www.youtube.com\/embed\/zkYhTQEUIUA\" title=\"Limits - Part - 3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 7:}\\  \\color {red} {Evaluate:}\\ \\lim\\ _{x\\ \\to\\ 3}\\ \\frac{x^5\\ -\\ 243}{x^2\\ -\\ 9}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  W.\\ K.\\ T\\ \\lim\\ _{x\\ \\to\\ a}\\ \\frac{x^n\\ -\\ a^n}{x\\ -\\ a}\\ =\\  n\\ a^{n\\ -\\ 1}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\lim\\ _{x\\ \\to\\ 3}\\ \\frac{x^5\\ -\\ 243}{x^2\\ -\\ 9}\\ =\\ \\lim\\ _{x\\ \\to\\ 3}\\ \\frac{x^5\\ -\\ 3^5}{x^2\\ -\\ 3^2} \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Multiply\\ and\\ divide\\ by\\ \u2018x &#8211; 3&#8217;\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\lim\\ _{x\\ \\to\\ 3}\\ \\frac{\\frac{x^5\\ -\\ 3^5}{x\\ -\\ 3}}{\\frac{x^2\\ -\\ 3^2}{x\\ -\\ 3}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5\\ 3^{5\\ -\\ 1}}{2\\ 3^{2\\ -\\ 1}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5\\ 3^4}{2\\ (3)}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5\\ (3^3)}{2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5\\ (27)}{2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{135}{2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- billboard -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:970px;height:250px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9933277607\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 8:}\\  \\color {Red} {Evaluate:}\\ \\lim\\ _{\u03b8\\ \\to\\ 0}\\ \\frac{Sin\\ 3\u03b8}{\u03b8}\\  \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  W.\\ K.\\ T\\ \\lim\\ _{\u03b8\\ \\to\\ 0}\\ \\frac{Sin\\ n\u03b8}{\u03b8}\\ =\\  n\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\lim\\ _{\u03b8\\ \\to\\ 0}\\ \\frac{Sin\\ 3\u03b8}{\u03b8}\\ =\\  3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 9.}\\ \\color {red} {Evaluate:}\\ \\lim\\ _{x\\ \\to\\ 0}\\ \\frac{Sin\\ 2x}{Sin\\ 4x}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ Feb\\ 2002,\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  W.\\ K.\\ T\\ \\lim\\ _{\u03b8\\ \\to\\ 0}\\ \\frac{Sin\\ n\u03b8}{\u03b8}\\ =\\  n\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Multiply\\ and\\ divide\\ by\\ \u2018x &#8216;\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\lim\\ _{x\\ \\to\\ 0}\\ \\frac{\\frac{Sin\\ 2x}{x}}{\\frac{Sin\\ 4x}{x}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{2}{4}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 10:}\\  \\color {Red} {Evaluate:}\\ \\lim\\ _{x\\ \\to\\ 0}\\ \\frac{Sin\\ 10x}{Sin\\ 7x}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  W.\\ K.\\ T\\ \\lim\\ _{\u03b8\\ \\to\\ 0}\\ \\frac{Sin\\ n\u03b8}{\u03b8}\\ =\\  n\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Multiply\\ and\\ divide\\ by\\ \u2018x &#8216;\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\lim\\ _{x\\ \\to\\ 0}\\ \\frac{\\frac{Sin\\ 10x}{x}}{\\frac{Sin\\ 7x}{x}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{10}{7}\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"853\" height=\"480\" src=\"https:\/\/www.youtube.com\/embed\/inA06Hwk09M\" title=\"Limits - Part - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 11:}\\  \\color {Red} {Evaluate:}\\ \\lim\\ _{\u03b8\\ \\to\\ 0}\\ \\frac{5Sin\\ 6\u03b8}{3Sin\\ 2\u03b8}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ June\\ 2002,\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  W.\\ K.\\ T\\ \\lim\\ _{\u03b8\\ \\to\\ 0}\\ \\frac{Sin\\ n\u03b8}{\u03b8}\\ =\\  n\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Multiply\\ and\\ divide\\ by\\ \u2018\u03b8 &#8216;\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\lim\\ _{\u03b8\\ \\to\\ 0}\\ \\frac{\\frac{5Sin\\ 6\u03b8}{\u03b8}}{\\frac{3Sin\\ 2\u03b8}{\u03b8}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5}{3}(\\frac{6}{2})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 5\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<h2 class=\"wp-block-heading has-text-align-center\">Exercise Problems<\/h2>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- A}}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"fluid\" data-ad-layout-key=\"-6x+db-3g-5s+t1\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"3292591486\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- B}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {1.}\\  \\color {red} {Evaluate:}\\ \\lim\\ _{x\\ \\to\\ a}\\ \\frac{\\sqrt{x}\\ -\\ \\sqrt{a}}{x\\ -\\ a}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {2.}\\ \\color {red} {Evaluate:}\\ \\lim\\ _{x\\ \\to\\ 0}\\ \\frac{Sin\\ 7x}{Sin\\ 3x}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Ad1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8240817448\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- C}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {3.}\\  \\color {red} {Evaluate:}\\ (i)\\ \\lim\\ _{x\\ \\to\\ 0}\\ \\frac{x^2\\ +\\ 7\\ x}{x^2\\ +\\ 5\\ x}\\ \\hspace{1cm}\\ (ii)\\ \\lim\\ _{\u03b8\\ \\to\\ 0}\\ \\frac{Sin\\ 3\\ \u03b8}{Sin\\ 2\\ \u03b8}\\ \\hspace{7cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {4.}\\  \\color {red} {Evaluate:}\\ (i)\\ \\lim\\ _{x\\ \\to\\ 3}\\ \\frac{x^6\\ -\\ 3^6}{x\\ -\\ 3}\\ \\hspace{1cm}\\ (ii)\\ \\lim\\ _{\u03b8\\ \\to\\ 0}\\ \\frac{Sin\\ 7\\ \u03b8}{Sin\\ 2\\ \u03b8}\\ \\hspace{7cm}\\] <\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Leader board 1 -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:728px;height:90px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8769628924\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n","protected":false},"excerpt":{"rendered":"<p>Exercise 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