{"id":1349,"date":"2020-03-22T05:40:50","date_gmt":"2020-03-22T05:40:50","guid":{"rendered":"http:\/\/ibalmaths.com\/?post_type=knowledgebase&#038;p=1349"},"modified":"2020-03-26T10:34:58","modified_gmt":"2020-03-26T10:34:58","slug":"ibdp-past-year-exam-questions-limits","status":"publish","type":"knowledgebase","link":"https:\/\/ibalmaths.com\/index.php\/ibdp-math-hl-2\/limits\/ibdp-past-year-exam-questions-limits\/","title":{"rendered":"IBDP Past Year Exam Questions &#8211; Limits"},"content":{"rendered":"<p><strong>1. [N18\/P3\/TZ0]<\/strong><\/p>\r\n<p>(a) Use L\u2019H\u00f4pital\u2019s rule to determine the value of<span style=\"font-family: 'times new roman', times, serif;\">\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><munder><mrow><mi>l<\/mi><mi>i<\/mi><mi>m<\/mi><\/mrow><mrow><mi>x<\/mi><mo>&#8594;<\/mo><mn>0<\/mn><\/mrow><\/munder><mfenced><mfrac><mrow><msup><mi>e<\/mi><mrow><mo>&#8211;<\/mo><mn>3<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><\/msup><mo>+<\/mo><mn>3<\/mn><mi>cos<\/mi><mfenced><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/mfenced><mo>&#8211;<\/mo><mo>&#160;<\/mo><mn>4<\/mn><\/mrow><mrow><mn>3<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><\/mfenced><\/math>\r\n . [5 marks]<\/span><\/p>\r\n<p>(b) Hence find\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><munder><mrow><mi>l<\/mi><mi>i<\/mi><mi>m<\/mi><\/mrow><mrow><mi>x<\/mi><mo>&#8594;<\/mo><mn>0<\/mn><\/mrow><\/munder><mfenced><mfrac><mrow><msubsup><mo>&#8747;<\/mo><mn>0<\/mn><mi>x<\/mi><\/msubsup><mfenced><mrow><msup><mi>e<\/mi><mrow><mo>&#8211;<\/mo><mn>3<\/mn><msup><mi>t<\/mi><mn>2<\/mn><\/msup><\/mrow><\/msup><mo>+<\/mo><mn>3<\/mn><mi>cos<\/mi><mfenced><mrow><mn>2<\/mn><mi>t<\/mi><\/mrow><\/mfenced><mo>&#8211;<\/mo><mo>&#160;<\/mo><mn>4<\/mn><\/mrow><\/mfenced><mo>d<\/mo><mi>t<\/mi><\/mrow><mrow><msubsup><mo>&#8747;<\/mo><mn>0<\/mn><mi>x<\/mi><\/msubsup><mn>3<\/mn><msup><mi>t<\/mi><mn>2<\/mn><\/msup><mo>d<\/mo><mi>t<\/mi><\/mrow><\/mfrac><\/mfenced><\/math>\r\n .\u00a0 \u00a0[3 marks]<\/p>\r\n<div id=\"link1-link-1349\" class=\"sh-link link1-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link1', 1349, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link1-toggle-1349\">Solution<\/span><\/a><\/div><div id=\"link1-content-1349\" class=\"sh-content link1-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1354\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-11.jpg\" alt=\"\" width=\"636\" height=\"600\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-11.jpg 636w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-11-300x283.jpg 300w\" sizes=\"auto, (max-width: 636px) 100vw, 636px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>2. [M17\/P3\/TZ0]<\/strong><\/p>\r\n<p>Use l\u2019H\u00f4pital\u2019s rule to determine the value of\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><munder><mrow><mi>l<\/mi><mi>i<\/mi><mi>m<\/mi><\/mrow><mrow><mi>x<\/mi><mo>&#8594;<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mi>x<\/mi><\/mrow><mrow><mi>x<\/mi><mi>ln<\/mi><mfenced><mrow><mn>1<\/mn><mo>+<\/mo><mi>x<\/mi><\/mrow><\/mfenced><\/mrow><\/mfrac><\/math>\r\n .\u00a0 [7 marks]<\/p>\r\n<div id=\"link2-link-1349\" class=\"sh-link link2-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link2', 1349, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link2-toggle-1349\">Solution<\/span><\/a><\/div><div id=\"link2-content-1349\" class=\"sh-content link2-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1357\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-9.jpg\" alt=\"\" width=\"1030\" height=\"482\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-9.jpg 1030w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-9-300x140.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-9-1024x479.jpg 1024w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-9-768x359.jpg 768w\" sizes=\"auto, (max-width: 1030px) 100vw, 1030px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>3. [N16\/P3\/TZ0]<\/strong><\/p>\r\n<p>Using l\u2019H\u00f4pital\u2019s rule, find\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><munder><mrow><mi>l<\/mi><mi>i<\/mi><mi>m<\/mi><\/mrow><mrow><mi>x<\/mi><mo>&#8594;<\/mo><mo>&#8734;<\/mo><\/mrow><\/munder><mfenced><mfrac><mrow><mi>a<\/mi><mi>r<\/mi><mi>c<\/mi><mi>sin<\/mi><mfenced><mstyle displaystyle=\"true\"><mfrac><mn>1<\/mn><msqrt><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/msqrt><\/mfrac><\/mstyle><\/mfenced><\/mrow><mstyle displaystyle=\"true\"><mfrac><mn>1<\/mn><msqrt><mi>x<\/mi><\/msqrt><\/mfrac><\/mstyle><\/mfrac><\/mfenced><\/math>\r\n .\u00a0 [6 marks]<\/p>\r\n<div id=\"link3-link-1349\" class=\"sh-link link3-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link3', 1349, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link3-toggle-1349\">Solution<\/span><\/a><\/div><div id=\"link3-content-1349\" class=\"sh-content link3-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1360\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-7.jpg\" alt=\"\" width=\"796\" height=\"628\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-7.jpg 796w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-7-300x237.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-7-768x606.jpg 768w\" sizes=\"auto, (max-width: 796px) 100vw, 796px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>4. [M16\/P3\/TZ0]<\/strong><\/p>\r\n<p>Determine the exact value of\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><munder><mrow><mi>l<\/mi><mi>i<\/mi><mi>m<\/mi><\/mrow><mrow><mi>x<\/mi><mo>&#8594;<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><msup><mi>e<\/mi><mi>x<\/mi><\/msup><mi>sin<\/mi><mi>x<\/mi><mo>&#8211;<\/mo><mo>&#160;<\/mo><mi>x<\/mi><mo>&#8211;<\/mo><mo>&#160;<\/mo><msup><mi>x<\/mi><mn>3<\/mn><\/msup><\/mrow><msup><mi>x<\/mi><mn>3<\/mn><\/msup><\/mfrac><\/math>\r\n .\u00a0 \u00a0[3 marks]<\/p>\r\n<div id=\"link4-link-1349\" class=\"sh-link link4-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link4', 1349, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link4-toggle-1349\">Solution<\/span><\/a><\/div><div id=\"link4-content-1349\" class=\"sh-content link4-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1362\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-6.jpg\" alt=\"\" width=\"952\" height=\"571\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-6.jpg 952w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-6-300x180.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-6-768x461.jpg 768w\" sizes=\"auto, (max-width: 952px) 100vw, 952px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>5. [M16\/P3\/TZ0]<\/strong><\/p>\r\n<p>Use l\u2019H\u00f4pital\u2019s rule to find\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><munder><mrow><mi>l<\/mi><mi>i<\/mi><mi>m<\/mi><\/mrow><mrow><mi>x<\/mi><mo>&#8594;<\/mo><mo>&#8734;<\/mo><\/mrow><\/munder><msup><mi>x<\/mi><mn>2<\/mn><\/msup><msup><mi>e<\/mi><mrow><mo>&#8211;<\/mo><mi>x<\/mi><\/mrow><\/msup><\/math>\r\n .\u00a0 [4 marks]<\/p>\r\n<div id=\"link5-link-1349\" class=\"sh-link link5-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link5', 1349, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link5-toggle-1349\">Solution<\/span><\/a><\/div><div id=\"link5-content-1349\" class=\"sh-content link5-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1365\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/5-5.jpg\" alt=\"\" width=\"1003\" height=\"143\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/5-5.jpg 1003w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/5-5-300x43.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/5-5-768x109.jpg 768w\" sizes=\"auto, (max-width: 1003px) 100vw, 1003px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>6. [M17]<\/strong><\/p>\r\n<p>Using l\u2019H\u00f4pital\u2019s rule, show that\u00a0\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><munder><mrow><mi>l<\/mi><mi>i<\/mi><mi>m<\/mi><\/mrow><mrow><mi>x<\/mi><mo>&#8594;<\/mo><mo>&#8734;<\/mo><\/mrow><\/munder><msup><mi>x<\/mi><mi>n<\/mi><\/msup><msup><mi>e<\/mi><mrow><mo>&#8211;<\/mo><mo>&#160;<\/mo><mi>x<\/mi><\/mrow><\/msup><mo>=<\/mo><mn>0<\/mn><\/math>\r\n where\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><mo>&#8712;<\/mo><mi mathvariant=\"normal\">&#8469;<\/mi><\/math>\r\n .\u00a0 [4 marks]<\/p>\r\n<div id=\"link6-link-1349\" class=\"sh-link link6-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link6', 1349, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link6-toggle-1349\">Solution<\/span><\/a><\/div><div id=\"link6-content-1349\" class=\"sh-content link6-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1366\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/6-9.jpg\" alt=\"\" width=\"949\" height=\"463\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/6-9.jpg 949w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/6-9-300x146.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/6-9-768x375.jpg 768w\" sizes=\"auto, (max-width: 949px) 100vw, 949px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>7. [M16]<\/strong><\/p>\r\n<p>Use l\u2019H\u00f4pital\u2019s rule to show that\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><munder><mrow><mi>l<\/mi><mi>i<\/mi><mi>m<\/mi><\/mrow><mrow><mi>x<\/mi><mo>&#8594;<\/mo><mo>&#8734;<\/mo><\/mrow><\/munder><mfrac><msup><mi>x<\/mi><mn>3<\/mn><\/msup><msup><mi>e<\/mi><mi>x<\/mi><\/msup><\/mfrac><mo>=<\/mo><mn>0<\/mn><\/math>\r\n .\u00a0 \u00a0[3 marks]<\/p>\r\n<div id=\"link7-link-1349\" class=\"sh-link link7-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link7', 1349, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link7-toggle-1349\">Solution<\/span><\/a><\/div><div id=\"link7-content-1349\" class=\"sh-content link7-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1373\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/7-6.jpg\" alt=\"\" width=\"997\" height=\"297\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/7-6.jpg 997w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/7-6-300x89.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/7-6-768x229.jpg 768w\" sizes=\"auto, (max-width: 997px) 100vw, 997px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>8. [M14]<\/strong><\/p>\r\n<p>Using l\u2019H\u00f4pital\u2019s rule, show that\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><munder><mrow><mi>l<\/mi><mi>i<\/mi><mi>m<\/mi><\/mrow><mrow><mi>x<\/mi><mo>&#8594;<\/mo><mo>&#8734;<\/mo><\/mrow><\/munder><mfrac><msup><mi>x<\/mi><mi>n<\/mi><\/msup><msup><mi>e<\/mi><mrow><mi>&#955;<\/mi><mi>x<\/mi><\/mrow><\/msup><\/mfrac><mo>=<\/mo><mn>0<\/mn><mo>;<\/mo><mo>&#160;<\/mo><mi>n<\/mi><mo>&#8712;<\/mo><msup><mi mathvariant=\"normal\">&#8484;<\/mi><mo>+<\/mo><\/msup><mo>&#160;<\/mo><mo>,<\/mo><mo>&#160;<\/mo><mi>&#955;<\/mi><mo>&#8712;<\/mo><msup><mi mathvariant=\"normal\">&#8477;<\/mi><mo>+<\/mo><\/msup><\/math>\r\n .<\/p>\r\n<div id=\"link8-link-1349\" class=\"sh-link link8-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link8', 1349, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link8-toggle-1349\">Solution<\/span><\/a><\/div><div id=\"link8-content-1349\" class=\"sh-content link8-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1375\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-6.jpg\" alt=\"\" width=\"1008\" height=\"691\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-6.jpg 1008w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-6-300x206.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/8-6-768x526.jpg 768w\" sizes=\"auto, (max-width: 1008px) 100vw, 1008px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>9. [M11\/P3\/TZ0]<\/strong><\/p>\r\n<p>(a) Find the first three terms of the Maclaurin series for \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>ln<\/mi><mfenced><mrow><mn>1<\/mn><mo>+<\/mo><msup><mi>e<\/mi><mi>x<\/mi><\/msup><\/mrow><\/mfenced><\/math>\r\n . [6 marks]<br \/>\r\n(b) Hence, or otherwise, determine the value of \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><munder><mrow><mi>l<\/mi><mi>i<\/mi><mi>m<\/mi><\/mrow><mrow><mi>x<\/mi><mo>&#8594;<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mn>2<\/mn><mi>ln<\/mi><mfenced><mrow><mn>1<\/mn><mo>+<\/mo><msup><mi>e<\/mi><mi>x<\/mi><\/msup><\/mrow><\/mfenced><mo>&#8211;<\/mo><mo>&#160;<\/mo><mi>x<\/mi><mo>&#8211;<\/mo><mo>&#160;<\/mo><mi>ln<\/mi><mn>4<\/mn><\/mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mfrac><\/math>\r\n \u00a0. [4 marks]<\/p>\r\n<div id=\"link9-link-1349\" class=\"sh-link link9-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link9', 1349, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link9-toggle-1349\">Solution<\/span><\/a><\/div><div id=\"link9-content-1349\" class=\"sh-content link9-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1377\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/9-9.jpg\" alt=\"\" width=\"953\" height=\"1248\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/9-9.jpg 953w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/9-9-229x300.jpg 229w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/9-9-782x1024.jpg 782w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/9-9-768x1006.jpg 768w\" sizes=\"auto, (max-width: 953px) 100vw, 953px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>10. [N11\/P3\/TZ0]<\/strong><\/p>\r\n<p>Find\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><munder><mrow><mi>l<\/mi><mi>i<\/mi><mi>m<\/mi><\/mrow><mrow><mi>x<\/mi><mo>&#8594;<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mrow><\/munder><mfenced><mfrac><mfenced><mstyle displaystyle=\"true\"><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mo>&#8211;<\/mo><mo>&#160;<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mstyle><\/mfenced><mrow><mi>c<\/mi><mi>o<\/mi><mi>t<\/mi><mo>&#160;<\/mo><mi>&#960;x<\/mi><\/mrow><\/mfrac><\/mfenced><mo>&#160;<\/mo><mo>&#160;<\/mo><mo>&#160;<\/mo><\/math>\r\n<\/p>\r\n<div id=\"link10-link-1349\" class=\"sh-link link10-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link10', 1349, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link10-toggle-1349\">Solution<\/span><\/a><\/div><div id=\"link10-content-1349\" class=\"sh-content link10-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1378\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/10-8.jpg\" alt=\"\" width=\"944\" height=\"262\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/10-8.jpg 944w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/10-8-300x83.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/10-8-768x213.jpg 768w\" sizes=\"auto, (max-width: 944px) 100vw, 944px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>11. [M09\/P3\/TZ0]<\/strong><\/p>\r\n<p>Find<\/p>\r\n<p>(i)\u00a0 \u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><munder><mrow><mi>l<\/mi><mi>i<\/mi><mi>m<\/mi><\/mrow><mrow><mi>x<\/mi><mo>&#8594;<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi>tan<\/mi><mi>x<\/mi><\/mrow><mrow><mi>x<\/mi><mo>&#160;<\/mo><mo>+<\/mo><mo>&#160;<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><\/math>\r\n ;\u00a0 \u00a0 [4 marks]<\/p>\r\n<p>(ii)\u00a0 \u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><munder><mrow><mi>l<\/mi><mi>i<\/mi><mi>m<\/mi><\/mrow><mrow><mi>x<\/mi><mo>&#8594;<\/mo><mn>1<\/mn><\/mrow><\/munder><mfrac><mrow><mn>1<\/mn><mo>&#160;<\/mo><mo>&#8211;<\/mo><mo>&#160;<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>&#160;<\/mo><mo>+<\/mo><mo>&#160;<\/mo><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mi>ln<\/mi><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><mo>&#160;<\/mo><mo>&#8211;<\/mo><mo>&#160;<\/mo><mi>sin<\/mi><mstyle displaystyle=\"true\"><mfrac><mi>&#960;x<\/mi><mn>2<\/mn><\/mfrac><\/mstyle><\/mrow><\/mfrac><\/math>\r\n .\u00a0 \u00a0 [7 marks]\u00a0\u00a0<\/p>\r\n<div id=\"link11-link-1349\" class=\"sh-link link11-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link11', 1349, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link11-toggle-1349\">Solution<\/span><\/a><\/div><div id=\"link11-content-1349\" class=\"sh-content link11-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1438\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/11-7.jpg\" alt=\"\" width=\"932\" height=\"507\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/11-7.jpg 932w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/11-7-300x163.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/11-7-768x418.jpg 768w\" sizes=\"auto, (max-width: 932px) 100vw, 932px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>12. [N10\/P3\/TZ0]<\/strong><\/p>\r\n<p>Find\u00a0\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><munder><mrow><mi>l<\/mi><mi>i<\/mi><mi>m<\/mi><\/mrow><mrow><mi>x<\/mi><mo>&#8594;<\/mo><mn>0<\/mn><\/mrow><\/munder><mfenced><mfrac><mrow><mn>1<\/mn><mo>&#8211;<\/mo><mi>cos<\/mi><msup><mi>x<\/mi><mn>6<\/mn><\/msup><\/mrow><msup><mi>x<\/mi><mn>12<\/mn><\/msup><\/mfrac><\/mfenced><\/math>\r\n.\u00a0 \u00a0 [7 marks]<\/p>\r\n<div id=\"link12-link-1349\" class=\"sh-link link12-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link12', 1349, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link12-toggle-1349\">Solution<\/span><\/a><\/div><div id=\"link12-content-1349\" class=\"sh-content link12-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1468\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12-5.jpg\" alt=\"\" width=\"960\" height=\"1292\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12-5.jpg 960w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12-5-223x300.jpg 223w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12-5-761x1024.jpg 761w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/12-5-768x1034.jpg 768w\" sizes=\"auto, (max-width: 960px) 100vw, 960px\" \/><\/p>\r\n<p>\u00a0<\/div><!-- AddThis Advanced Settings generic via filter on the_content --><!-- AddThis Share Buttons generic via filter on the_content -->","protected":false},"excerpt":{"rendered":"1. [N18\/P3\/TZ0] (a) Use L\u2019H\u00f4pital\u2019s rule to determine the value of\u00a0 limx&#8594;0e&#8211;3&#215;2+3cos2x&#8211;&#160;43&#215;2 . [5 marks] (b) Hence find\u00a0 limx&#8594;0&#8747;0xe&#8211;3t2+3cos2t&#8211;&#160;4dt&#8747;0x3t2dt .\u00a0 \u00a0[3 marks] 2. [M17\/P3\/TZ0] Use l\u2019H\u00f4pital\u2019s rule to determine the value of\u00a0 limx&#8594;0sin2xxln1+x .\u00a0 [7 marks] 3. [N16\/P3\/TZ0] Using l\u2019H\u00f4pital\u2019s rule, find\u00a0 limx&#8594;&#8734;arcsin1x+11x .\u00a0 [6 marks] 4. [M16\/P3\/TZ0] Determine the exact value of\u00a0 limx&#8594;0exsinx&#8211;&#160;x&#8211;&#160;x3x3 .\u00a0 [&hellip;]<!-- AddThis Advanced Settings generic via filter on get_the_excerpt --><!-- AddThis Share Buttons generic via filter on get_the_excerpt -->","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-1349","knowledgebase","type-knowledgebase","status-publish","hentry","knowledgebase_cat-limits","no-wpautop"],"_links":{"self":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1349","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase"}],"about":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/types\/knowledgebase"}],"author":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/comments?post=1349"}],"version-history":[{"count":21,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1349\/revisions"}],"predecessor-version":[{"id":1469,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1349\/revisions\/1469"}],"wp:attachment":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/media?parent=1349"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}